CHAPTER 6 Time Value of Money

Transcription

6-1
Chapter 6
The Time Value
of Money
Future Value
Present Value
Rates of Return
Amortization
Copyright © 2000 by Harcourt, Inc.
All rights reserved. Requests for permission
to make copies of any part of the work should
be mailed to the following address:
Permissions Department, Harcourt, Inc., 6277
Sea Harbor Drive, Orlando, Florida 328876777.
Copyright (C) 2000 by Harcourt, Inc. All rights
reserved.
6-2
Cash Flow Time Lines
Graphical representations used to
show the timing of cash flows.
0
1
2
CF1
CF2
3
i%
CF0
CF3
Tick marks at ends of periods, so t=0 is today;
t=1 is the end of Period 1; or the beginning of
Period 2.
reserved.
6-3
Time line for a
$100 lump sum due
at the end of Year 2.
0
1
2
Year
i%
100
reserved.
6-4
Time line for an ordinary
annuity of $100 for 3 years.
0
1
2
100
100
3
i%
100
reserved.
6-5
Time line for uneven CFs
-$50 at t=0 and $100, $75, and $50
at the end of Years 1 through 3.
0
1
2
100
75
3
i%
-50
50
reserved.
6-6
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.
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?
FVn = FV1 = PV + INT
= PV + (PV x i)
= PV (1 + i)
= $100(1+0.05) = $100(1.05) = $105
reserved.
6-7
What’s the FV of an initial $100
after 3 years if i = 10%?
0
1
2
3
i=10%
100
FV = ?
Finding FVs is Compounding.
reserved.
6-8
Future Value
After 1 year:
FV1 =
PV + i = PV + PV (i)
=
PV(1 + i)
=
$100 (1.10)
=
$110.00.
After 2 years:
FV2 =
PV(1 + i)2
=
$100 (1.10)2
=
$121.00.
After 3 years:
FV3 =
PV(1 + i)3
=
100 (1.10)3
=
$133.10.
In general, FVn = PV (1 + i)n
reserved.
6-9
Three ways to find
Future Values:
 Solve the Equation with a
Regular Calculator
 Use Tables
 Use a Financial Calculator
reserved.
6-10
Financial Calculator Solution:
Financial calculators solve this equation:
FV n PV 1  i .
n
There are 4 variables. If 3 are known,
the calculator will solve for the 4th.
reserved.
6-11
Here’s the setup to find FV:
INPUTS
3
N
10
-100
I/YR PV
OUTPUT
0
PMT
FV
133.10
Clearing automatically sets everything to
0, but for safety enter PMT = 0.
Set: P/YR = 1
reserved.
6-12
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 cash flow or
series of cash flows, the reverse of
compounding.
reserved.
6-13
What is the PV of $100 due
in 3 years if i = 10%?
0
1
2
3
10%
PV = ?
100
reserved.
6-14
What is the PV of $100 due
in 3 years if i = 10%?
Solve FVn = PV (1 + i )n for PV:
n
FV
1 

n
PV =
= FV 

n
 1+ i 
1 + in
3
1

 = $100 PVIF 
PV = $100 

i,n 
1.10 
= $100 0.7513  = $75.13.
reserved.
6-15
INPUTS
OUTPUT
3
N
10
I/YR
PV
-75.13
0
PMT
100
FV
Either PV or FV must be negative. Here
PV = -75.13. Put in $75.13 today, take
out $100 after 3 years.
reserved.
6-16
If sales grow at 20% per year,
how long before sales double?
Solve for n:
FVn = 1(1 + i)n;
2 = 1(1.20)n .
reserved.
6-17
INPUTS
OUTPUT
Graphical
Illustration: 2
20
I/YR
N
3.8
-1
PV
0
PMT
2
FV
FV
3.8
1
Year
0
1
2
3
4
reserved.
6-18
Future Value of an Annuity
 Annuity: A series of payments of an
equal amount 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.
reserved.
6-19
Ordinary Annuity
Versus
Annuity Due
0
i%
0
1
2
PMT
PMT
1
2
PMT
PMT
3
PMT
3
i%
PMT
reserved.
6-20
What’s the FV of a 3-year
Ordinary Annuity of $100 at 10%?
0
1
2
100
100
3
10%
100
110
121
FV
= 331
reserved.
6-21
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
331.00
reserved.
6-22
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
reserved.
6-23
What is the PV of this
Ordinary Annuity?
0
1
2
100
100
3
10%
100
90.91
82.64
75.13
248.69 = PV
reserved.
6-24
INPUTS
3
N
OUTPUT
10
I/YR
100
PV
PMT
0
FV
-248.69
Have payments but no lump sum FV,
so enter 0 for future value.
reserved.
6-25
Find the FV and PV if the
Annuity were an Annuity Due.
0
1
2
100
100
3
10%
100
reserved.
6-26
Switch from “End” to “Begin”.
Then enter variables to find PVA3 = $273.55.
INPUTS
OUTPUT
3
10
N
I/YR
100
PV
PMT
0
FV
-273.55
Then enter PV = 0 and press FV to find
FV = $364.10.
reserved.
6-27
Solving for Interest Rates
with Annuities
You pay $864.80 for an investment that promises to
pay you $250 per year for the next four years, with
payments made at the end of the year. What interest
rate will you earn on this investment?
0
1
2
3
4
250
250
250
i=?
- 864.80
250
reserved.
6-28
PVAn
= PMT(PVIFAi,n)
$846.80
= $250(PVIFA i = ?,4)
INPUTS
OUTPUT
4
?
N
I/YR
- 846.80 250
PV
PMT
0
FV
=7.0
reserved.
6-29
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
 Cash flow (CF) designates cash flows
in general, including uneven cash
flows
reserved.
6-30
What is the PV of this
Uneven Cash Flow Stream?
0
1
2
3
4
100
300
300
-50
10%
90.91
247.93
225.39
-34.15
530.08 = PV
reserved.
6-31
 Input in “Cash Flow” register:
CF0 =
0
CF1 =
100
CF2 =
300
CF3 =
300
CF4 =
-50
 Enter I = 10%, then press NPV button to
get NPV = 530.09. (Here NPV = PV.)
reserved.
6-32
What interest rate would cause
$100 to grow to $125.97 in 3 years?
$100 (1 + i )3 = $125.97.
INPUTS
3
N
OUTPUT
-100
I/YR
PV
0
PMT
125.97
FV
8%
reserved.
6-33
Semiannual and Other
Compounding Periods
 Annual compounding is the arithmetic
process of determining the final value of
a cash flow or series of cash flows when
interest is added once a year.
 Semiannual compounding is the
arithmetic process of determining the
final value of a cash flow or series of
cash flows when interest is added twice a
year.
reserved.
6-34
Will the FV of a lump sum
be larger or smaller if we
compound more often,
holding the state i% constant?
Why?
LARGER! If compounding is more
frequent than once a year--for example,
semi-annually, quarterly, or daily-interest is earned on interest more often.
reserved.
6-35
0
1
2
3
10%
100
133.10
Annually: FV3 = 100(1.10)3 = 133.10.
0
1
0
1
2
2
3
4
3
5
6
5%
100
134.01
Semi-annually: FV6/2 = 100(1.05)6 = 134.01.
reserved.
6-36
Distinguishing Between
Different Interest Rates
iSIMPLE = Simple (Quoted) Rate
used to compute the interest paid per period
EAR = Effective Annual Rate
the annual rate of interest actually being earned
APR = Annual Percentage Rate
periodic rate X the number of periods per year
reserved.
6-37
How do we find EAR for a
simple rate of 10%,
compounded semi-annually?
m
i

EAR = 1 + SIMPLE - 1

m 
2
0.10

 - 1.0
=  1+
2 
2
= 1.05 - 1.0
= 0.1025 = 10.25%.
reserved.
6-38
FV of $100 after 3 years
under 10% semi-annual
compounding? Quarterly?
iSIMPLE 

FVn = PV  1 +

m 

FV3s
mxn
0.10

= $100 1 +


2 
2x3
= $100(1.05)6 = $134.01
FV3Q = $100(1.025)12 = $134.49
reserved.
6-39
Fractional Time Periods
Example: $100 deposited in a bank
at 10% interest for 0.75 of the year
0
0.25
0.50
0.75
1.00
8%
- 100
INPUTS
FV = ?
0.75
N
OUTPUT
10
I/YR
- 100
PV
0
PMT
?
FV
=107.41
reserved.
6-40
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, etc.
They are very important!
 Financial calculators (and spreadsheets)
are great for setting up amortization
tables.
reserved.
6-41
Construct an amortization schedule
for a $1,000, 10% annual rate loan
with 3 equal payments.
Step 1: Find the required payments
0
1
2
PMT
PMT
3
10%
-1000
INPUTS
OUTPUT
3
10
-1000
N
I/YR
PV
PMT
0
PMT
FV
402.11
reserved.
6-42
Step 2: Find interest charge for Year 1
INTt = Beg balt (i)
INT1 = 1000(0.10) = $100.
Step 3: Find repayment of principal in Year 1
Repmt. = PMT - INT
= 402.11 - 100
= $302.11.
Step 4: Find ending balance after Year 1
End bal = Beg bal - Repmt
= 1000 - 302.11 = $697.89.
Repeat these steps for Years 2 and 3
to complete the amortization table.
reserved.
6-43
Loan Amortization Table
10 Percent Interest Rate
YR
Beg Bal
PMT
1
$1000
$402
$100
$302
$698
2
698
402
70
332
366
3
366
402
37
366
0
1,206.34
206.34
1,000
Total
INT Prin PMT End Bal
Interest declines. Tax Implications.
reserved.
6-44
$
402.11
Interest
302.11
Principal Payments
0
1
2
3
Level payments. Interest declines because
outstanding balance declines. Lender earns
10% on loan outstanding, which is falling.
reserved.
6-45
Comparison of Different
Types of Interest Rates
 iSIMPLE:
Written into contracts, quoted by banks
and brokers. Not used in calculations or shown on
time lines.
 iper:
Used in calculations, shown on time lines.
If iSIMPLE has annual compounding,
then iper = iSIMPLE/1 = iSIMPLE.
 EAR : Used to compare returns on investments
with different payments per year.
(Used for calculations if and only if dealing with annuities where
payments don’t match interest compounding periods.)
reserved.
6-46
Simple (Quoted) Rate
 iSIMPLE is stated in contracts.
Periods per year (m) must also be given.
 Examples:
8%; Quarterly
8%, Daily interest (365 days)
reserved.
6-47
Periodic Rate
 Periodic rate = iper = iSIMPLE/m, where m is
periods per year. m = 4 for quarterly, 12
for monthly, and 360 or 365 for daily
compounding.
 Examples:
8% quarterly: iper = 8/4 = 2%
8% daily (365): iper = 8/365 = 0.021918%
reserved.
6-48
Effective Annual Rate
Effective Annual Rate:
The annual rate which cause PV to grow to the same
FV as under multiperiod compounding.
Example: EAR for 10%, semiannual:
FV
EAR
=
=
=
(1 + iSIMPLE/m)m
(1.05)2 = 1.1025.
10.25% because
(1.1025)1 = 1.1025.
Any PV would grow to same FV at 10.25% annually or
10% semiannually.
reserved.
6-49
End of Chapter 6
The Time Value
of Money
reserved.

CHAPTER 6 Time Value of Money

Transcription

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