CHAPTER 2
Discounted Cash Flow Analysis
Time Value of Money
Financial Mathematics
Future value
Present value
Rates of return
Amortization
Annuities, AND
Many Examples
2-1
2-2
MINICASE 2
SIMPLE?
p. 88
Also note financial mathematics problems at end of TAB &
Notes on Excel and LOTUS.
2-3
MINICASE 2
Why is financial mathematics
(time value of money) so
important in financial
analysis?
2-4
a.Time lines show timing of cash flows.
ALWAYS A GOOD IDEA TO DRAW A TIME LINE.
0
CF0
i%
1
2
CF1
CF2
3
CF3
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; and so on.
2-5
Time line for a $100 lump sum due at
the end of Year 2.
0
1
2 Years
i%
100
2-6
Time line for an ordinary annuity of
$100 for 3 years.
0
i%
1
2
3
100
100
100
2-7
Time line for uneven CFs -$50 at t = 0
and $100, $75, and $50 at the end of
Years 1 through 3.
0
-50
i%
1
2
3
100
75
50
2-8
b(1) What’s the FV of an initial
$100 after 3 years if i = 10%?
0
100
10%
1
2
3
FV = ?
Finding FVs is compounding.
2-9
b(1) What’s the FV of an initial
$100 after 3 years if i = 10%?
0
100
10%
1
110
2
?
3
FV = ?
Finding FVs is compounding.
2 - 10
After 1 year
FV1 = PV + INT1 = PV + PV(i)
= PV(1 + i)
= $100(1.10)
= $110.00.
After 2 years
FV2 = FV1(1 + i)
= PV(1 + i)2
= $100(1.10)2
= $121.00.
2 - 11
After 3 years
FV3 = PV(1 + i)3
= 100(1.10)3
= $133.10.
In general,
FVn = PV(1 + i)n.
2 - 12
Four Ways to Find FVs
 Solve the equation with a regular
calculator
 Use tables
 Use a financial calculator
 Use a spreadsheet
2 - 13
USING TABLES: See handout
3 PERIODS
10 %
= 1.3310
times 100 = $133.10
SAY GOOD-BYE TO USING TABLES!
2 - 14
Financial Calculator Solution
Financial calculators solve this
equation:
F V n  P V  1  i .
n
There are 4 variables. If 3 are
known, the calculator will solve for
the 4th.
2 - 15
Here’s the setup to find FV:
INPUTS
3
N
10
I/YR
-100
PV
0
PMT
OUTPUT
FV
133.10
Clearing automatically sets
everything to 0, but for safety enter
PMT = 0.
Set: P/YR = 1, END
2 - 16
b(2) 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
PV = ?
10%
1
2
3
100
2 - 17
Solve FVn = PV(1 + i )n for PV:
n
FVn
PV = (1 + i)n
3
( )
PV = $100/ 1.1
=
0
= $100(0.7513)
= $75.13.
2 - 18
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.
2 - 19
EXCEL SOLUTION
LOOK AT FUNCTION’S PAGE FOR
EXCEL/LOTUS.
2 - 20
Spreadsheet Solution
Use the FV function: see spreadsheet
in Ch 02 Mini Case.xls.
 = FV(Rate, Nper, Pmt, PV)
 = FV(0.10, 3, 0, -100) = 133.10
2 - 21
Spreadsheet Solution
Use the PV function: see spreadsheet.
 = PV(Rate, Nper, Pmt, FV)
 = PV(0.10, 3, 0, 100) = -75.13
2 - 22
c. If sales grow at 20% per year, how
long before sales double?
Solve for n:
FVn
2
(1.20)n
n ln (1.20)
n(0.1823)
n
Time line ?
= PV(1 + i)n
= 1(1.20)n
=2
= ln 2
= 0.6931
= 0.6931/0.1823 = 3.8 years.
2 - 23
INPUTS
OUTPUT
N
3.8
20
I/YR
-1
PV
0
PMT
2
FV
Beware:Some Calculators round up.
Graphical Illustration:
FV
2
3.8
1
Years
0
1
2
3
4
2 - 24
Spreadsheet Solution
Use the NPER function: see
spreadsheet.
= NPER(Rate, Pmt, PV, FV)
 = NPER(0.20, 0, -1, 2) = 3.8
Correction
2 - 25
ADDITIONAL QUESTION
A FARMER CAN SPEND $60/ACRE
TO PLANT PINE TREES ON SOME
MARGINAL LAND. THE EXPECTED
REAL RATE OF RETURN IS 4%, AND
THE EXPECTED INFLATION RATE IS
6%. WHAT IS THE EXPECTED
VALUE OF THE TIMBER AFTER 20
YEARS?
2 - 26
ADDITIONAL QUESTION
Bill Veeck once bought the Chicago
White Sox for $10 million and then
sold it five years later for $20 million.
In short, he doubled his money in
five years. What compound rate of
return did Veeck earn on his
investment?
2 - 27
RULE OF 72
A good approximation of the interest
rate--or number of years--required to
double your money.
n * krequired to double = 72
In this case,
5 * krequired to double = 72
• k = 14.4
 Correct answer was 14.87, so for ball-park
approximation, use Rule of 72.
2 - 28
ADDITIONAL QUESTION
John Jacob Astor bought an acre of
land in Eastside Manhattan in 1790
for $58. If average interest rate is
5%, did he make a good deal?
2 - 29
d. What’s the difference between an
ordinary annuity and an annuity due?
2 - 30
d. What’s the difference between an
ordinary annuity and an annuity due?
Ordinary Annuity
0
1
2
PMT
PMT
1
2
PMT
PMT
3
i%
PMT
Annuity Due
0
3
i%
PMT
36
2 - 31
HINT
ANNUITY DUE OF n PERIODS IS
EQUAL TO A REGULAR ANNUITY OF
(n-1) PERIODS PLUS THE PMT.
2 - 32
e(1). What’s the FV of a 3-year ordinary
annuity of $100 at 10%?
0
1
2
3
100
100
100
10%
FV =
2 - 33
e(1). 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
2 - 34
FV Annuity Formula
The future value of an annuity with n
periods and an interest rate of i can
be found with the following formula:
n
 PMT
(1  i)  1
i
3
 100
(1  0 . 10 )  1
0.10
 331 .
2 - 35
Financial Calculator Formula
for Annuities
Financial calculators solve this
equation:
FV
n
 PV 1 i





n





n 1
(1

i)
 PMT
 0.
i
There are 5 variables. If 4 are
known, the calculator will solve
for the 5th.
Correct but confusing!
2 - 36
Financial Calculator Solution
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
331.00
Have payments but no lump sum PV,
so enter 0 for present value.
2 - 37
Spreadsheet Solution
Use the FV function: see spreadsheet.
 = FV(Rate, Nper, Pmt, Pv)
 = FV(0.10, 3, -100, 0) = 331.00
2 - 38
e(2). What’s the PV of this ordinary annuity?
0
10%
_____
= PV
1
2
3
100
100
100
2 - 39
What’s the PV of this ordinary annuity?
0
10%
90.91
82.64
75.13
248.69 = PV
1
2
3
100
100
100
2 - 40
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-248.69
Have payments but no lump sum FV,
so enter 0 for future value.
2 - 41
Spreadsheet Solution
Use the PV function: see spreadsheet.
 = PV(Rate, Nper, Pmt, Fv)
 = PV(0.10, 3, 100, 0) = -248.69
2 - 42
e(3). Find the FV and PV if the
annuity were an annuity due.
0
100
10%
1
2
100
100
3
2 - 43
Could, on the 12C, switch from
“End” to “Begin”; i.e. f 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.
2 - 44
Another HINT
FV OF AN ANNUITY DUE OF n
PERIODS IS EQUAL TO THE FV OF A
REGULAR ANNUITY OF n PERIODS
TIMES (1+k)
PV OF AN ANNUITY DUE OF n
PERIODS IS EQUAL TO THE PV OF A
REGULAR ANNUITY OF n PERIODS
TIMES (1+k)
(slide 30)
2 - 45
HINT, illlustrated
The PV of this regular annuity was
248.69.
Multiply this by (1 + .10), and you get:
273.55, the PV of the annuity due.
This avoids the necessity of having
to switch from end to begin.
2 - 46
PV and FV of Annuity Due
vs. Ordinary Annuity
PV of annuity due:
 = (PV of ordinary annuity) (1+i)
= (248.69) (1+ 0.10) = 273.56
FV of annuity due:
= (FV of ordinary annuity) (1+i)
= (331.00) (1+ 0.10) = 364.1
2 - 47
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.
2 - 48
Excel Function for Annuities Due
Change the formula to:
=PV(10%,3,-100,0,1)
The fourth term, 0, tells the function
there are no other cash flows. The
fifth term tells the function that it is an
annuity due. A similar function gives
the future value of an annuity due:
=FV(10%,3,-100,0,1)
2 - 49
EXCEL SOLUTION
2 - 50
(f) What is the PV of this uneven cash
flow stream?
0
1
2
3
4
100
300
300
-50
10%
______
= PV
2 - 51
(f) What is the PV of this uneven cash
flow stream?
0
10%
1
2
3
4
100
300
300
-50
90.91
247.93
225.39
-34.15
530.08 = PV
2 - 52
Input in “CFLO” register:
CF0 =
0
CF1 = 100
CF2 = 300
CF3 = 300
CF4 = -50
Enter I = 10%, then press NPV button
to get NPV = 530.09.
2 - 53
CALCULATOR SOLUTION
0
100
300
2
50
10
f
g
CF0
g
CFj
g
CFj
g
Nj
CHS g CFj
i
NPV
2 - 54
EXCEL SOLUTION
REMEMBER EXCEL/LOTUS READS
THE 1ST CASH FLOW AS OCCURING
ONE PERIOD HENCE.
2 - 55
Spreadsheet Solution
1
A
B
C
D
E
0
1
2
3
4
100
300
300
-50
2
3
530.09
Excel Formula in cell A3:
=NPV(10%,B2:E2)
2 - 56
g. 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
8.00%
PV
0
PMT
125.97
FV
2 - 57
EXCEL SOLUTION
2 - 58
h.Will the FV of a lump sum be larger
or smaller if we compound more often,
holding the stated i% constant? Why?
2 - 59
h.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, semiannually, quarterly,
or daily--interest is earned on
interest more often.
2 - 60
0
1
2
3
10%
100
133.10
Annually: FV3 = 100(1.10)3 = 133.10.
0
0
1
1
2
3
2
4
5
3
6
5%
100
134.01
Semiannually: FV6 = 100(1.05)6 = 134.01.
2 - 61
Periodic rate = iPer = iNom/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: iPer = 8/4 = 2%.
8% daily (365): iPer = 8/365 = 0.021918%.
2 - 62
 Effective Annual Rate (EAR = EFF%):
The annual rate which causes PV to
grow to the same FV as under
multiperiod compounding.
2 - 63
An investment with monthly
compounding is different from
one with quarterly compounding.
Must put on EAR% basis to
compare rates of return.
Banks say “interest paid daily.”
Same as compounded daily.
2 - 64
h(3).How do we find EAR for a
nominal rate of 10%, compounded
semiannually?
EFF%

1

=
+
i Nom 

m 
m
- 1
2
=
=
=
0 .1 0 

 1+
 - 1 .0

2 
2
 1 .0 5  - 1 .0
0 .1 0 2 5 = 1 0 .2 5 % .
Or use a financial calculator (not 12C)
EAR = (1+knom/m)m - 1
.10
ENT
2
divide
1
+
2
Yx
1
=
.1025
(1 + EAR) = (1+knom/m)m
2 - 65
2 - 66
CALCULATOR
WHAT IS EAR IF COMPOUNDING
QUARTERLY?
COMPOUNDING DAILY?
COMPOUNDING CONTINUOUSLY?
2 - 67
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(360) = (1 + 0.10/360)360 - 1 = 10.52%.
2 - 68
For multiple years, n
(1 + EAR) = (1 + Knom/m)m
(1 + EAR)n = (1 + Knom/m)mn
To multiply by a $ of dollars, PRIN
PRIN * (1 + EAR)n = PRIN * (1 + Knom/m)mn
2 - 69
i. 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.
2 - 70
When is each rate used?
iNom:
Written into contracts, quoted
by banks and brokers. May be
used in calculations or shown
on time lines when
compounding is annual.
OR USE EAR!
2 - 71
iPer:
Used in calculations,
shown on time lines.
If iNom has annual compounding,
then iPer = iNom/1 = iNom.
Can always use EAR!
2 - 72
EAR = EFF%: Used to compare
returns on investments
with different
compounding patterns.
Also used for calculations if dealing
with annuities where payments
don’t match interest compounding
periods.
2 - 73
FV of $100 after 3 years under 10%
semiannual compounding? Quarterly?
Daily?
iNom

FVn = PV 1 +


m
FV3S
FV3Q
mn
0.10

= $100 1 +


2 
.
2x3
= $100(1.05)6 = $134.01.
= $100(1.025)12 = $134.49.
2 - 74
ALTERNATE SOLUTION USING EAR
FOR SEMIANNUAL COMPOUNDING,
EAR = 10.25%
FOR 3 YEARS: 100*(1.1025)3 =
$134.01
FOR Quarterly COMPOUNDING and 3
years:
100*(1.1038)3 = $134.49
2 - 75
j(3). 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%
100
100
5
6
6-mos.
periods
100
2 - 76
Payments occur annually, but
compounding occurs each 6
months.
So we can’t use normal annuity
valuation techniques.
2 - 77
1st Method: Compound Each CF
0
5%
1
2
3
4
6
100.00
110.25
121.55
331.80
FVA3 = 100(1.05)4 + 100(1.05)2 + 100
= 331.80.
100
100
5
2 - 78
2nd Method: Treat as an Annuity
I.E. USE EAR
Could you find FV with a
financial calculator?
Yes, by following these steps:
a. Find the EAR for the quoted rate:
EAR =
(
0.10
1+ 2
2
) - 1 = 10.25%.
2 - 79
Or, to find EAR with a 17 OR 19b Calculator:
NOM%
P/YR
EFF%
=
=
=
10
2
10.25
2 - 80
b. The cash flow stream is an annual
annuity whose EFF% = 10.25%.
c.
INPUTS
OUTPUT
3
10.25
N
I/YR
0
-100
PV
PMT
FV
331.80
2 - 81
j(2) What’s the PV of this stream?
0
5%
1
2
3
100
100
100
2 - 82
What’s the PV of this stream?
0
5%
90.70
82.27
74.62
247.59
1
2
3
100
100
100
2 - 83
Calculator solution
100 PMT
3
n
10.25 i
f
NPV
=
247.59
2 - 84
What’s the FV of this stream under
quarterly compouning?
0
1
2
3
100
100
100
2 - 85
EAR WORKSHEET
EAR worksheet
knom =
m=
EAR =
0.12 INPUT
8 INPUT
0.126493
EAR=
m=
knom=
0.1255 INPUT
4 INPUT
0.119992
2 - 86
k. Amortization
Construct an amortization schedule
for a $1,000, 10% annual rate loan
with 3 equal payments.
2 - 87
Step 1: Find the required payments.
0
10%
-1000
INPUTS
OUTPUT
1
2
3
PMT
PMT
PMT
3
10
-1000
N
I/YR
PV
0
PMT
402.11
FV
2 - 88
ALGEBRA
PMT/(1+k) + PMT/(1+k)2 + PMT/(1+k)3
= 1000
PMT [1/(1+k) + 1/(1+k)2 + 1/(1+k)3] =
1000, or
PMT =
1000/ [1/(1+k) + 1/(1+k)2 + 1/(1+k)3]
2 - 89
Step 2: Find interest charge for Year 1.
INTt = Beg balt (i)
INT1 = 1,000(0.10) = $100.
Step 3: Find repayment of principal in
Year 1.
Repmt = PMT - INT
= 402.11 - 100
= $302.11.
2 - 90
Step 4: Find ending balance after
Year 1.
End bal
= Beg bal - Repmt
= 1,000 - 302.11 = $697.89.
Repeat these steps for Years 2 and 3
to complete the amortization table.
2 - 91
YR
BEG
BAL
PRIN
PMT
INT
1
$1,000
$402
$100
2
698
402
70
3
366
402
37
TOT
1,206.34 206.34
REDUCTION
$302
332
366
1,000
END
BAL
$698
366
0
Interest declines. Tax implications.
2 - 92
$
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.
2 - 93
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.
2 - 94
EXCEL SOLUTION
2 - 95
NEW PROBLEM:
On January 1 you deposit $100 in an
account that pays a nominal interest
rate of 10%, with daily compounding
(365 days).
How much will you have on October
1, or after 9 months (273 days)?
(Days given.)
2 - 96
iPer
0
= 10.0% / 365
= 0.027397% per day.
1
2
273
0.027397%
FV=?
-100
FV273 = $1001.00027397 
= $1001.07765  = $107.77.
273
Note: % in calculator, decimal in equation.
2 - 97
iPer = iNom/m
= 10.0/365
= 0.027397% per day.
INPUTS
273
N
OUTPUT
-100
I/YR
PV
0
FV
PMT
107.77
Enter i in one step.
Leave data in calculator.
2 - 98
Now suppose you leave your money
in the bank for 21 months, which is
1.75 years or 273 + 365 = 638 days.
How much will be in your account at
maturity?
Answer: Override N = 273 with N =
638. FV = $119.10.
2 - 99
iPer = 0.027397% per day.
0
-100
365
638 days
FV = 119.10
FV = $100(1 +
0.10/365)638
= $100(1.00027397)638
= $100(1.1910)
= $119.10.
2 - 100
ALTERNATIVE SOLUTION USING EAR
FIND EAR
.10
ENTER
365
DIVIDE
1
+
365
Yx
[= (1 + EAR)]
.75
Yx
[= (1 + EAR).75]
100
MULTIPLY
 = $107.79
2 - 101
MORE PRECISELY, instead of .75
exponent:
Calculate [1+EAR] as above, then
273 ENTER
365
Yx
DIVIDE
[=(1+EAR)]
[=(1+EAR)(273/365)]
100
MULTIPLY
= $107.77
2 - 102
PROBLEM
SUPPOSE THAT YOU WERE TOLD
THAT THE EFFECTIVE ANNUAL
RATE WERE 10%, WITH DAILY
COMPOUNDING. WHAT THE
STATED, OR NOMINAL RATE BE IN
THIS CASE?
2 - 103
ALGEBRA
(1 + EAR) = (1 + knom /m)m
(1 + EAR)(1/m) = (1 + knom /m)
(1 + EAR)(1/m) - 1 = knom /m
m*[(1 + EAR)(1/m) - 1] = knom
2 - 104
m*[(1 + EAR)(1/m) - 1] = knom
Using the calculator, EAR = 10%, daily
compounding.
1.1
365
1
365
=
ENTER
1/X Yx
MULTIPLY
9.53%
2 - 105
n. You are offered a note which pays
$1,000 in 15 months (or 456 days) for
$850. You have $850 in a bank
which pays a 7.0% nominal rate, with
365 daily compounding, which is a
daily rate of 0.019178% and an EAR
of 7.25%. You plan to leave the
money in the bank if you don’t buy
the note. The note is riskless.
Should you buy it?
2 - 106
iPer =0.019178% per day.
0
-850
365
456 days
1,000
3 Ways to Solve:
1. Greatest future wealth: FV
2. Greatest wealth today: PV
3. Highest rate of return: Highest EFF%
2 - 107
1. Greatest Future Wealth
Find FV of $850 left in bank for
15 months and compare with
note’s FV = $1000.
FVBank
=
=
$850(1.00019178)456
$927.67 in bank.
Buy the note: $1000 > $927.67.
2 - 108
Calculator Solution to FV:
iPer = iNom/m
= 7.0/365
= 0.019178% per day.
INPUTS
456
N
I/YR
-850
0
PV
PMT
OUTPUT
Enter iPer in one step.
FV
927.67
2 - 109
2. Greatest Present Wealth
Find PV of note, and compare
with its $850 cost:
PV = $1000/[(1.00019178)456]
= $916.27.
2 - 110
INPUTS
7/365 =
456 .019178
N
OUTPUT
I/YR
0
PV
1000
PMT
FV
-916.27
PV of note is greater than its $850
cost, so buy the note. Raises your
wealth.
2 - 111
3. Rate of Return
Find the EFF% on note and
compare with 7.25% bank pays,
which is your opportunity cost of
capital:
FVn = PV(1 + i)n
1000 = $850(1 + i)456
Now we must solve for i.
2 - 112
INPUTS
456
N
OUTPUT
-850
I/YR
PV
0.035646%
per day
0
1000
PMT
FV
Convert % to decimal:
Decimal = 0.035646/100 = 0.00035646.
EAR = EFF% = (1.00035646)365 - 1
= 13.89%.
2 - 113
Using interest conversion:
P/YR
NOM%
EFF%
=
=
=
365
0.035646(365) = 13.01
13.89
Since 13.89% > 7.25% opportunity cost,
buy the note.
2 - 114
ADDITIONAL PROBLEM #2
IT IS NOW JANUARY 1. YOU PLAN
TO MAKE 5 DEPOSITS OF $100
EACH, ONE EVERY 6 MONTHS, WITH
THE FIRST PAYMENT BEING MADE
TODAY. IF THE BANK PAYS A
NOMINAL INTEREST RATE OF 12
PERCENT, SEMIANNUAL
COMPOUNDING, HOW MUCH WILL
BE IN YOUR ACCOUNT AFTER 10
YEARS?
2 - 115
ADDITIONAL PROBLEM #3
 IT IS NOW JANUARY 1, 1997. YOU ARE
OFFERED A NOTE UNDER WHICH
SOMEONE PROMISES TO MAKE 5
PAYMENTS OF $100 EACH, WITH THE
FIRST PAYMENT ON JULY 1, 1997 AND
SUBSEQUENT PAYMENTS ON EACH JULY
1 THEREAFTER THROUGH JULY 1, 2001,
PLUS A FINAL PAYMENT OF $1000 TO BE
MADE ON JANUARY 1, 2002. WITH A
NOMINAL DISCOUNT RATE OF 10
PERCENT, QUARTERLY COMPOUNDING,
WHAT IS THE PV OF THE NOTE?
2 - 116
JOHNM PROBLEMS