CHAPTER 5
Time Value of Money
The most powerful force in the universe is compound interest-Albert
Einstein

Future value
Concept/Math/Using calculator

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Present value
Concept/Math/Calculator/solving for N,i


Annuities
Concept/math/calculator/other variables
perpetuity

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Uneven cash flow
Amortization
Simple case
Real Life case: excel

6-1
What is the future value (FV) of an initial
$100 after 3 years, if I/YR = 10%?

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
FV: The amount to which a cash flow or series of cash flows
will grow over a give period of time when compounded at a
given interest rate.
Finding the FV of a cash flow or series of cash flows when
compound interest is applied is called compounding.
FV can be solved by using the arithmetic, financial calculator,
and spreadsheet methods.
0
1
2
3
10%
100
FV = ?
6-2
Solving for FV:
The arithmetic method




After 1 year:
 FV1 = PV ( 1 + i ) = $100 (1.10)
= $110.00
After 2 years:
2
2
 FV2 = PV ( 1 + i ) = $100 (1.10)
=$121.00
After 3 years:
3
3
 FV3 = PV ( 1 + i ) = $100 (1.10)
=$133.10
After n years (general case):
n
 FVn = PV ( 1 + i )
n
 PV= FVn /( 1 + i )
6-3
Solving for FV:
The calculator method


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Calculator settings
Solves the general FV equation.
Requires 4 inputs into calculator, and will
solve for the fifth. (Set to P/YR = 1 and
END mode.)
INPUTS
OUTPUT
3
10
-100
0
N
I/YR
PV
PMT
FV
133.10
6-4
What is the present value (PV) of $100
due in 3 years, if I/YR = 10%?


PV: The value today of a future cash flow or
series of cash flows
Finding the PV of a cash flow or series of
cash flows when compound interest is
applied is called discounting (the reverse of
compounding).
0
1
2
3
10%
PV = ?
100
6-5
Solving for PV:
The arithmetic method

Solve the general FV equation for PV:


PV = FVn / ( 1 + i )n
PV = FV3 / ( 1 + i )3
= $100 / ( 1.10 )3
= $75.13
6-6
Solving for PV:
The calculator method


Solves the general FV equation for PV.
Exactly like solving for FV, except we
have different input information and are
solving for a different variable.
INPUTS
OUTPUT
3
10
N
I/YR
PV
0
100
PMT
FV
-75.13
6-7
Solving for N:
If interest is 20% per year, how long
before your savings double?


Solves the general FV equation for N.
Same as previous problems, but now
solving for N.
INPUTS
N
OUTPUT
20
-1
0
2
I/YR
PV
PMT
FV
3.8
6-8
Solving for I:
What interest rate would cause $100 to
grow to $125.97 in 3 years?

Solves the general FV equation for I.
INPUTS
3
N
OUTPUT
I/YR
-100
0
125.97
PV
PMT
FV
8
6-9
Annuity: A series of payments of an
equal amount at fixed intervals for a
specified number of periods
Ordinary Annuity
0
i%
1
2
3
PMT
PMT
PMT
6-10
Solving for FV of annuity:
3-year ordinary annuity of $100 at 10%

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Timeline and formula:
Using calculator
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
331
6-11
Solving for PV:
3-year ordinary annuity of $100 at 10%

$100 payments still occur at the end of
each period, but now there is no FV.
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-248.69
6-12
Application of annuity
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
Suppose you are 60, expect to live for
another 20 years. What is the present
value of an annuity with a annual
payment of $100,000, assuming a 8%
annual interest rate?
Saving for retirement
Computing PMT, I, and N
Combining annuity with lump sum
6-13
Perpetuity

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Perpetuity: A stream of equal payments
expected to continue forever
PV of Perpetuity = PMT/I
6-14
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
6-15
Using calculator (quick guide p13)
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CF mode
Clear previous work: 2nd+CLR WORK
Two major steps
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Input CF and frequency
Press NPV, input I, arrow down,then CPT

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Note: consecutive cash flows of the same amount can be
entered as one cash flow with frequency higher than one
Example: Quick guide page 13
6-16
Solving for PV:
Uneven cash flow stream
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Input cash flows in the calculator’s “CFLO”
register:
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CF0
CF1
CF2
CF3
CF4
=
=
=
=
=
0
100
300
300
-50
press NPV, Enter I/YR = 10, arrow down, press
CPT to get NPV = $530.09. (Here NPV = PV.)
6-17
Loan amortization
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Amortization tables are widely used for
home mortgages, auto loans, business
loans, retirement plans, etc.
Financial calculators and spreadsheets are
great for setting up amortization tables.
EXAMPLE: Construct an amortization
schedule for a $1,000, 10% annual rate
loan with 3 equal payments.
6-18
Step 1:
Find the required annual payment

All input information is already given,
just remember that the FV = 0 because
the reason for amortizing the loan and
making payments is to retire the loan.
INPUTS
OUTPUT
3
10
-1000
N
I/YR
PV
0
PMT
FV
402.11
6-19
Step 2:
Find the interest paid in Year 1

The borrower will owe interest upon the
initial balance at the end of the first
year. Interest to be paid in the first
year can be found by multiplying the
beginning balance by the interest rate.
INTt = Beg balt (i)
INT1 = $1,000 (0.10) = $100
6-20
Step 3:
Find the principal repaid in Year 1

If a payment of $402.11 was made at
the end of the first year and $100 was
paid toward interest, the remaining
value must represent the amount of
principal repaid.
PRIN= PMT – INT
= $402.11 - $100 = $302.11
6-21
Step 4:
Find the ending balance after Year 1

To find the balance at the end of the
period, subtract the amount paid
toward principal from the beginning
balance.
END BAL = BEG BAL – PRIN
= $1,000 - $302.11
= $697.89
6-22
Constructing an amortization table:
Repeat steps 1 – 4 until end of loan
Year
BEG BAL PMT
INT
PRIN
END
BAL
$302
$698
1
$1,000
$402
$100
2
698
402
70
332
366
3
366
402
37
366
0
1,206.34
206.34
1,000
-
TOTAL

Interest paid declines with each payment as
the balance declines. What are the tax
implications of this?
6-23
Illustrating an amortized payment:
Where does the money go?
$
402.11
Interest
302.11
Principal Payments
0
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1
2
Constant payments.
Declining interest payments.
Declining balance.
3
6-24
Power of compounding
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The most powerful force in universe:
Compounding
Compare total return at
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Low rate, inflation rate, stock market
return rate and high rate
Short vs. Long horizon
6-25
Power of compounding
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Investing is not a hit-and-run. Investing
for the long run!
Time has value and time is on your
side.
The snow ball effect
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Long slope
Steeper slope
Consistent slope
6-26
Using TV table
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(optional)
FV of Lump Sum
PV of Lump Sum
FV of annuity
PV of annuity
6-27