6-1
CHAPTER 5
Time Value of Money





Read Chapter 6 (Ch. 5
in the 4th edition)
Future value
Present value
Rates of return
Amortization
6-2
Time Value of Money Problems




Use a financial calculator
Bring your calculator to class
Will need on exams
We will not use the tables
6-3

Time lines show timing of cash
flows.
0
1
2
CF1
CF2
3
i%
CF0

Tick marks at ends of periods, so
Time 0 is today; Time 1 is the end
of Period 1; or the beginning of
Period 2.
CF3
6-4
A. (1) a. Time line for a
$100 lump sum due at the
end of Year 2.
0
1
2
i%
100
Year
6-5
A. (1) b. Time line for an
ordinary annuity of $100 for
3 years.
0
1
2
100
100
3
i%
100
6-6
A. (1) c. 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
6-7
What’s the FV of an initial
$100 after 3 years if i = 10%?
0
1
2
3
10%
100
FV = ?
Finding FVs is Compounding.
6-8
After 1 year:
FV1 =
=
=
=
PV + I1
=
PV(1 + i)
$100 (1.10)
$110.00.
After 2 years:
FV2 =
=
=
PV(1 + i)2
$100 (1.10)2
$121.00.
PV + PV (i)
6-9
After 3 years:
FV3 =
=
=
PV(1 + i)3
100 (1.10)3
$133.10.
In general,
FVn =
PV (1 + i)n
6-10
Three ways to find FVs:
1.
2.
3.
4.
‘Solve’ the Equation with a
Scientific Calculator
Use Tables (the book describes
this but not for use in this class)
Use a Financial Calculator
Spreadsheet (has built-in
formulas) -- won’t work on exams
6-11
Here’s the setup to find FV:
INPUTS
OUTPUT
3
N
10
-100
I/YR PV
0
PMT
FV
133.10
Clearing automatically sets everything to
0, but for safety enter PMT = 0.
Check your calculator. Set: P/YR = 1 and
END (“BEGIN” should not show on the display)
6-12
What’s the PV of $100 due
in 3 years if i = 10%?
Finding PVs is discounting,
and it’s the reverse of
compounding.
0
1
2
3
10%
PV = ?
100
6-13
Financial Calculator Solution:
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.
6-14
If sales grow at 20% per year,
how long before sales double?
Solve for n:
FVn = 1(1 + i)n; In our case
2 = (1.20)n .
Take the log of both sides:
ln(2) = n ln(1.2)
n = ln(2)/ln(1.2)=.693…/0.1823..
=3.8017
6-15
Financial calculator solution
INPUTS
OUTPUT
N
3.8
20
I/YR
-1
PV
0
PMT
2
FV
Graphical Illustration:
FV
2
3.8
1
Year
0
1
2
3
4
6-16
What’s the difference
between an ordinary
annuity and an annuity
due?
6-17
Ordinary vs. Annuity Due
0
i%
0
1
2
PMT
PMT
1
2
PMT
PMT
i%
PMT
3
PMT
3
6-18
What’s the FV of a 3-year
ordinary annuity of $100 at
10%?
0
1
2
3
100
100
100
10%
110
121
FV
= 331
6-19
Financial Calculator Solution:
INPUTS
OUTPUT
3
10
0
N
I/YR
PV
-100
PMT
FV
331.00
If you enter PMT of 100, you get FV of
-331.
Get used to the fact that you have to figure
out the sign.
6-20
What’s the PV of this ordinary
annuity?
0
1
2
100
100
3
10%
90.91
82.64
75.13
248.69 = PV
100
6-21
Financial Calculator Solution:
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.
6-22
Technical Aside:
Your calculator really is assuming a NPV
equation, with PV as a time zero cash flow
as follows:
1  (1  i)n 
n
NPV  PV  PMT 

FV
(
1

i
)

i


When you use the top row of calculator keys,
the calculator assumes NPV=0 and solves for
one variable.
6-23
Find the FV and PV if the
annuity were an annuity due.
0
1
2
100
100
10%
100
3
6-24
Switch from “End” to “Begin”.
Then enter variables to find PVA3 = $273.55.
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-273.55
Then enter PV = 0 and press FV to find
FV = $364.10.
Alternative:
6-25
The first payment is in the present and
thus has a PV of 100.
n The next two payments comprise a two
period ordinary annuity -- use the formula
with n=2, PMT=100, and i=.10.
n Sum the above two for the present value.
n If you already have the PV, multiply by (1  i)3
To get FV
n
6-26
Perpetuities
n
n
n
A perpetuity is a stream of regular payments that
goes on forever
An infinite annuity
Future value of a perpetuity
Makes no sense because there is no end point
Present value of a perpetuity
A diminishing series of numbers
• Each payment’s present value is smaller
than the one before
PVp 
PMT
k
6-27
Perpetuities—Example
Example
Q: The Longhorn Corporation issues a security that promises to pay its holder $5
per quarter indefinitely. Money markets are such that investors can earn about
8% compounded quarterly on their money. How much can Longhorn sell this
special security for?
A: Convert the k to a quarterly k and plug the values into the equation.
PVp 
PMT
$5

 $250
k
0.02
You may also work this by inputting a
large n into your calculator (to
simulate infinity), as shown below.
N
999
I/Y
2
PMT
5
FV
0
PV
250
Answer
6-28
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
6-29
n
n
Input in “CFLO” register ( CFj ):
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.)
6-30
What’s Project L’s NPV?
Project L:
0
1
2
10
60
3
10%
-100.00
9.09
80
1
1.1
49.59
60.11
18.79 = NPVL
2
1 .1
1.13
6-31
Calculator Solution:
Enter in CFLO for L:
-100
CF0
10
CF1
60
CF2
80
CF3
10
i
NPV
= 18.78 = NPVL
6-32
TI Calculators
•BA-35 doesn’t appear to do uneven cash
flows (NPV and IRR)
BA II PLUS
CF
CF0=
-100
Enter

C01=
10
Enter

F01=
1.00 
C02=
60
Enter

F02=
1.00 
C03=
NPV
80
Enter
IRR
I=10
CPT
IRR=
Enter
 F03=
1.00
 CPT NPV=
18.78
18.13
6-33
The Sinking Fund Problem
n
Companies borrow money by issuing bonds for
lengthy time periods
No repayment of principal is made during the
bonds’ lives
• Principal is repaid at maturity in a lump sum
– A sinking fund provides cash to pay off a
bond’s principal at maturity
• Problem is to determine the periodic
deposit to have the needed amount at the
bond’s maturity—a future value of an
annuity problem
Example
The Sinking Fund Problem –Example
6-34
Q: The Greenville Company issued bonds totaling $15 million for 30 years. The
bond agreement specifies that a sinking fund must be maintained after 10
years, which will retire the bonds at maturity. Although no one can accurately
predict interest rates, Greenville’s bank has estimated that a yield of 6% on
deposited funds is realistic for long-term planning. How much should Greenville
plan to deposit each year to be able to retire the bonds with the money put
aside?
A: The time period of the annuity is the last 20 years of the bond issue’s life. Input
the following keystrokes into your calculator.
N
20
I/Y
6
FV
15,000,000
PV
0
PMT
407,768.35
Answer
6-35
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
8%
PV
0
PMT
125.97
FV
Will the FV of a lump sum be
larger or smaller if we
compound more often, holding
the stated 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.
6-36
6-37
0
1
2
3
10%
100
133.10
Annually: FV3 = 100(1.10)3 = 133.10.
Semi-annually:
0
0
1
1
2
2
3
4
3
5
6
5%
100
FV6/2 = 100(1.05)6 = 134.01.
134.01
6-38
We will deal with 3
different rates:
iNom = nominal, or stated, or
quoted, rate per year.
iPer = periodic rate. The literal rate
applied each period
EAR = EFF% = effective
annual rate.
6-39
 iNom is stated in contracts. Periods per
year (m) must also be given. Sometimes
(incorrectly) referred to as the “simple”
interest rate.
 Examples:
• 8%, Daily interest (365 days)
• 8%; Quarterly
6-40


Periodic rate = iPer = iNom/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%
6-41

Effective Annual Rate (EAR = EFF%):
The annual rate which cause PV to grow
to the same FV as under multiperiod
compounding.
Example: EFF% for 10%, semiannual:
FV =
(1 + inom/m)m
=
(1.05)2 = 1.1025.
Any PV would grow to same FV at 10.25%
annually or 10% semiannually:
(1.1025)1 = 1.1025
(1.05)2 = 1.1025
6-42
Comparing Financial Investments


An investment with monthly
payments is different from
one with quarterly payments.
Must put on EFF% basis to
compare rates of return. Use
EFF% only for comparisons.
Banks say “interest paid
daily.” Same as compounded
daily.
6-43
How do we find EFF% for a
nominal rate of 10%, compounded
semi-annually?
m
i
EFF% =  1 + nom  - 1

m
2
0.10

=  1+
 - 1.0

2 
2
= 1.05 - 1.0
= 0.1025 = 10.25%.
6-44
EAR = EFF% of 10%
EARAnnual
=
10%.
EARQ = (1 + 0.10/4)4 - 1
=
10.38%.
EARM = (1 + 0.10/12)12 - 1 =
10.47%.
EARD = (1 + 0.10/360)360 - 1= 10.5155572%.
Continuous : e
.10
 1.105170918
6-45
Can the effective rate ever be
equal to the nominal rate?


Yes, but only if annual
compounding is used, i.e., if m = 1.
If m > 1, EFF% will always be
greater than the nominal rate.
6-46
When is each rate used?
inom:
Written into contracts,
quoted by banks and
brokers. Not used in
calculations or shown
on time lines.
6-47
iper:
Used in calculations,
shown on time lines.
If inom has annual compounding,
then iper = inom/1 = inom.
6-48
EAR = EFF%: Used to compare returns
on investments with different payments
per year and in advertising of deposit interest
rates.
(Used for calculations if and only if
dealing with annuities where payments
don’t match interest compounding
periods.)
6-49
FV of $100 after 3 years
under 10% semi-annual
compounding? Quarterly?
inom 

FVn = PV 1 +


m
FV3s
mn
0.10

= $100 1 +


2 
2x3
= $100(1.05)6 = $134.01
FV3Q = $100(1.025)12 = $134.49
What’s the value at the end
of Year 3 of the following CF
stream if the quoted interest
rate is 10%, compounded
semi-annually?
0
1
2
3
4
5
6-50
6
5%
100
100
6-month periods
100
6-51


Payments occur annually,
but compounding occurs
each 6 months.
So we can’t use normal
annuity valuation
techniques.
6-52
1st method: Compound each CF
0
1
2
4
3
5
6
5%
100
100
2
100(1.05)
100(1.05)
4
FVA3 = 100(1.05)4 + 100(1.05)2 + 100
= 331.80
100.00
110.25
121.55
331.80
6-53
What’s the PV of this stream?
1
2
3
100
100
100
0
5%
90.70
82.27
74.62
247.59
100 (1 . 05 )
2
100 (1 . 05 )
4
100 (1 . 05 )
6
Years
6-54
Second Method: use your
financial calculator!
Follow these two steps:
a.
Find the EAR for the quoted rate:
2
0.10

EAR =  1 +
 - 1 = 10.25%.

2 
This is the iper for a period of one
year. Use in formula (or calculator)
with the period equal to a year.
6-55
Time line
0
1
2
3
100
100
100
10.25%
6-56
b. Calculator inputs
INPUTS
OUTPUT
3
10.25
0
-100
N
I/YR
PV
PMT
FV
331.80
Calculator Workout: fill in the blanks
N
I
10
10
5
8
7
15
PV
100
PMT
FV
0
0
100
-500
100
0
-750
100
1000
240
8/12
-100,000
50
10
100
0
10
6-57
6-58
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
10%
- 100
INPUTS
FV = ?
0.75
N
OUTPUT
10
I/YR
- 100
PV
0
PMT
?
FV
=107.41
6-59
AMORTIZATION
Construct an amortization schedule
for a $1,000, 10% annual rate loan
with 3 equal payments.
6-60
This is what an amortization schedule looks like.
Amortization Table
Beginning
Principal
Period
1
Balance
Ending
Total
Interest Principal Principal
Payment Payment Payment Balance
$1,000.00 $402.11 $100.00 $302.11
$697.89
2
$697.89
$402.11
$69.79
$332.33
$365.56
3
$365.56
$402.11
$36.56
$365.56
$0.00
6-61
Step 1: Find the required payment.
0
1
2
3
PMT
PMT
10%
-1000
INPUTS
OUTPUT
3
10
-1000
N
I/YR
PV
PMT
0
PMT
402.11
FV
6-62
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.
6-63
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.
6-64
Amortization Table
Beginning
Principal
Period
1
Balance
Ending
Total
Interest Principal Principal
Payment Payment Payment Balance
$1,000.00 $402.11 $100.00 $302.11
$697.89
2
$697.89
$402.11
$69.79
$332.33
$365.56
3
$365.56
$402.11
$36.56
$365.56
$0.00
Interest declines. Tax Implications.
6-65


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.
6-66
Amortized Loans—Example
Example
Q: Suppose you borrow $10,000 over four years at 18% compounded
monthly repayable in monthly installments. How much is your loan
payment?
A: Adjust your interest rate and number of periods for monthly
compounding and input the following keystrokes into your calculator.
N
48
I/Y
1.5
PV
10,000
FV
0
PMT
293.75
Answer
This can also be calculated
using the PVA formula of
PVA = PMT[PVFAk, n] with
an n of 48 and a k of 1.5%,
resulting in $10,000 =
PMT[34.0426] = $293.75.
6-67
Amortized Loans—Example
Example
Q: Suppose you want to buy a car and can afford to make payments of
$500 a month. The bank makes three-year car loans at 12%
compounded monthly. How much can you borrow toward a new car?
A: Adjust your k and n for monthly compounding and input the following
calculator keystrokes.
N
36
I/Y
1
FV
0
PMT
500
PV
15,053.75
Answer
This can also be calculated
using the PVA formula of
PVA = PMT[PVFAk, n] with
an n of 36 and a k of 1%,
resulting in PVA =
$500[30.1075] = $15,053.75.
6-68
Loan Amortization Schedules—Example
Example
Q: Develop an amortization schedule for the
loan demonstrated in Example 5.12.
Note that the Interest portion
of the payment is decreasing
while the Principal portion is
increasing.