2.
Compound Interest
Simple interest (interest is earned only on the
original principal)
Compound interest (interest is earned on
principal and on interest received)
Year Beginning Balance Interest Earned Ending Amount
1 $100 $10 $110
2 $110 $11 $121
3 $121 $12 $133
4 $133 $13 $146
5 $146 $15 $161
Total Interest $61
4
Effects of Compounding
5
Cash Flow Time Lines
Graphical representations used to
show timing of cash flows
Time: 0 1 2 3
6%
FV = ?
Cash Flows: PV=100
Time 0 is today
Time 1 is the end of Period 1, etc.
6
2
3.
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.
7
Terms Used in Calculating Future Value
PV = Present value, or the beginning amount
that is invested
r = Interest rate paid on the account each
period
FVn = Future value of the account at the end
of n periods
n = number of period interest is earned
8
Future Value
In general, FVn = PV (1 + r)n
9
3
4.
Four Ways to Solve Time Value of
Money Problems
Use Cash Flow Time Line
Use Equations
Use Financial Calculator
Use Electronic Spreadsheet
10
Time Line Solution
The Future Value of $100 invested at 10% per
year for 3 years
Time(t): 0 1 2 3
10%
x (1.10) x (1.10) x (1.10)
Account balance: 100.00 110.00 121.00 133.10
11
Numerical (Equation) Solution
The Future Value of $100 invested at 10% per year for 3 years
FVn = PV(1+ r) n
FV3 =$100(1.10)3
=$100(1.331)
=133.10
12
4
5.
Financial Calculator Solution
The Future Value of $100 invested at 10% per year for 3 years
INPUTS 3 10 -100 0 ?
N I/YR PV PMT FV
OUTPUT 133.10
13
Future Value Relationships
Please note that the graph should start at $1
14
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.
15
5
6.
What is the PV of $100 due in 3 years
if r = 10%?
Time(t): 0 1 2 3
10%
x (1.10) x (1.10) x (1.10)
Account balance: 75.13 82.64 90.90 100.00
16
What is the PV of $100 due in 3 years
if r = 10%?
⎡ 1 ⎤
PV = FVn ⎢ n⎥
⎣ (1+ r) ⎦
⎡ 1 ⎤
PV = $100 ⎢ 3⎥
⎣(1.10) ⎦
= $100(0.7513)= $75.13
17
What interest rate would cause $100
to grow to $125.97 in 3 years?
0 1 2 3
r=?
100 (1 + r )3 = 125.97
PV = 100 ] 100 = 125.97/ (1 + r )3 [
125.97 = FV
18
6
7.
What interest rate would cause $100
to grow to $125.97 in 3 years?
$100 (1 + r )3 = $125.97
INPUTS 3 ? -100 0 125.97
N I/YR PV PMT FV
OUTPUT
8%
19
How many years will it take for
$68.30 to grow to $100
at an interest rate of 10%?
0 1 2 n-1 n=?
10%
...
68.30 (1.10 )n = 100.00
PV = 68.30 ] 68.30 = $100.00 /(1.10 )n [100.00 = FV
20
How many years will it take for
$68.30 to grow to $100 if interest of
10% is paid each year?
FVn = PV (1 + r ) n
$100.00 = $68.30(1.10) n
$100
(1.10) n = = 1.46413
$68.30
ln[(1.10) n ] = n[ln(1.10)] = ln(1.46413)
ln(1.46413)
n= = 4 .0
ln(1.10)
21
7
8.
How many years will it take for
$68.30 to grow to $100 if interest of
10% is paid each year?
INPUTS ? 10 -68.30 0 100.00
N I/YR PV PMT FV
OUTPUT
4.0
22
Present Value Relationships
23
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.
24
8
9.
Ordinary Annuity Versus Annuity Due
Ordinary Annuity
0 1 2 3
r%
PMT PMT PMT
Annuity Due
0 1 2 3
r%
PMT PMT PMT
25
What’s the FV of a 3-year Ordinary
Annuity of $100 at 5%?
0 1 2 3
5%
100 100 100
105
110.25
FV = 315.25
26
Numerical Solution:
⎡n−1 ⎤ ⎡ (1 + r) n −1⎤
FVA n = PMT⎢∑ (1 + r) n ⎥ = PMT⎢ ⎥
⎣t = 0 ⎦ ⎣ r ⎦
⎡ (1.05)3 −1⎤
FVA3 = $100⎢ ⎥
⎣ 0.05 ⎦
= $100(3.15250) = $315.25
27
9
11.
Financial Calculator Solution
Switch from “End” to “Begin”.
INPUTS 3 5 0 -100 ?
N I/YR PV PMT FV
OUTPUT 331.01
31
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.
32
What is the PV of this Ordinary
Annuity?
0 1 2 3
10%
100 100 100
90.91
82.64
75.13
248.69 = PV
33
11
13.
Financial Calculator Solution
INPUTS 3 5 ? -100 0
N I/YR PV PMT FV
OUTPUT 285.94
37
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 each year.
What interest rate will you earn on this
investment?
0 1 2 3 4
r=?
- 864.80 250 250 250 250
38
Numerical Solution
Use trial-and-error by substituting different values
of r into the following equation until the right side
equals $864.80.
⎡1 - 1 4 ⎤
$864.80 = $250⎢ (1+r) ⎥
⎢ r
⎣ ⎥
⎦
39
13
14.
Financial Calculator Solution
INPUTS 4 ? -846.80 250 0
N I/YR PV PMT FV
OUTPUT 7.0
40
Perpetuities
Annuities that go on indefinitely
Payment PMT
PVP = =
Interest rate r
41
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.
42
14
16.
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.
46
Will the FV of a lump sum be larger or
smaller if we compound more often,
holding the stated r constant? Why?
LARGER!
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.
47
Compounding
Annually vs. Semi-Annually
0 1 2 3
10%
100
133.10
Annually: FV3 = 100(1.10)3 = 133.10
0 1 2 3 4 5 6
5%
100 134.01
Semi-annually: FV6/2 = 100(1.05)6 = 134.01
48
16
17.
Distinguishing Between
Different Interest Rates
rSIMPLE = Simple (Quoted) Rate
used to compute the interest paid per period
rEAR = Effective Annual Rate
the annual rate of interest actually being
earned
APR = Annual Percentage Rate = rSIMPLE
periodic rate X the number of periods per year
49
Simple (Quoted) Rate
rSIMPLE is stated in contracts
Periods per year (m) must also be given
Examples:
8%, compounded quarterly
8%, compounded daily (365 days)
50
Periodic Rate
Periodic rate = rPER = rSIMPLE/m
Where m is number of compounding periods
per year. m = 4 for quarterly, 12 for monthly,
and 360 or 365 for daily compounding.
Examples:
8% quarterly: rPER = 8/4 = 2%
8% daily (365): rPER = 8/365 = 0.021918%
51
17
18.
Effective Annual Rate
The annual rate that causes PV to grow to the same
FV as under multi-period compounding.
Example: 10%, compounded semiannually:
rEAR = (1 + rSIMPLE/m)m - 1.0
= (1.05)2 - 1.0 = 0.1025 = 10.25%
52
How do we find rEAR for a simple rate
of 10%, compounded semi-annually?
m
⎛ r ⎞
rEAR = ⎜ 1 + SIMPLE ⎟ -1
⎝ m ⎠
2
⎛ 0.10 ⎞
= ⎜1 + ⎟ - 1.0
⎝ 2 ⎠
= (1.05 ) - 1.0 = 0.1025 = 10.25%
2
53
FV of $100 after 3 years if interest is
10% compounded semi-annual?
Quarterly?
⎛ r ⎞m×n
FVn = PV⎜1 + SIMPLE ⎟
⎝ m ⎠
2×3
⎛ 0.10 ⎞
FV3×2 = $100⎜1 + ⎟ = $100(1.340 10) = $134.01
⎝ 2 ⎠
4×3
⎛ 0.10 ⎞
FV3×4 = $100⎜1 + ⎟ = $100(1.344 89) = $134.49
⎝ 4 ⎠
54
18
19.
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
55
Step 1: Determine the required
payments
0 1 2 3
10%
-1000 PMT PMT PMT
INPUTS 3 10 -1000 ? 0
N I/YR PV PMT FV
OUTPUT 402.11
56
Step 2: Find simple interest charge for
Year 1
INTt = Beginning balancet (r)
INT1 = 1,000(0.10) = $100.00
57
19
20.
Loan Amortization Table
10 Percent Interest Rate
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 36.56 365.55 0.02
Total 1,206.33 206.35 999.98 *
* Rounding difference
58
Chapter 4 Essentials
Why is it important to understand and apply time
value to money concepts?
To be able to compare various investments
What is the difference between a present value
amount and a future value amount?
Future value adds interest - present value subtracts interest
What is an annuity?
A series of equal payments that occur at equal time intervals
59
Chapter 4 Essentials
What is the difference between the Annual
Percentage Rate and the Effective Annual Rate?
APR is a simple interest rate quoted on loans. EAR is the
actual interest rate or rate of return.
What is an amortized loan?
A loan paid off in equal payments over a specified period
How is the return on an investment determined?
The amount to which the investment will grow in the future
minus the cost of the investment
60
20
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