principles of finance 5e, 9 the time value of money © 2012 cengage learning. all rights reserved....
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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.
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.
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
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.
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
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.
4
Cash Flow Time Lines
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.
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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.
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.
6
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?
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FVn = FV1 = PV + INT
= PV + PV(r)
= PV (1 + r)
= $100(1 + 0.05) = $100(1.05) = $105
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.
What’s the FV of an initial $700 after three years if r = 10%?
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.
8
FV = ?
0 1 2 310%
700
Finding FV is Compounding
Future Value
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.
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
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.
10
Numerical (Equation) Solution
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.
11
PV = $700, r = 10%, and n =3
Financial Calculator Solution
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.
12
There are 4 variables. If 3 are known, the calculator will solve for the 4th.
Financial Calculator Solution
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.
13
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
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.
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?
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.
15
Solve for n:
FVn = 1(1 + r)n
2 = 1(1.20)n
The numerical solution is somewhat difficult.
Graphical Illustration
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.
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0 1 2 3 4
1
2
3.8
Year
FV
Financial Calculator Solution
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.
17
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.
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.
18
Ordinary Annuity Versus Annuity Due
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.
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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%
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.
20
Numerical Solution
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.
21
Financial Calculator Solution
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.
22
INPUTS
OUTPUT
3 5 0 -400 ?
=1261.00
N I/Y PV PMT FV
Spreadsheet Solution
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.
23
FV of a 3-year Annuity Due of $400 at 5%
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.
24
Numerical Solution
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.
25
Financial Calculator Solution
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.
26
INPUTS
OUTPUT
3 5 0 -400 ?
=1,324.05
N I/Y PV PMT FV
BGN
Spreadsheet Solution
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.
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
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.
28
Numerical Solution
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.
29
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.
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.
30
What is the PV of $935 due in three years if r = 10%?
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.
31
Numerical Solution
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.
32
Solve FVn = PV (1 + r )n for PV:
Financial Calculator Solution
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.
33
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.
Spreadsheet Solution
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.
34
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
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.
35
What is the PV of this Ordinary Annuity?
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.
36
Numerical Solution
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.
37
Financial Calculator Solution
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.
38
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
Spreadsheet Solution
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.
39
Present Value of an Annuity Due
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.
40
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)