CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics

1
CHAPTER 2Discounted Cash Flow AnalysisTime
Value of MoneyFinancial Mathematics

• Future value
• Present value
• Rates of return
• Amortization
• Annuities, AND
• Many Examples

2
MINICASE 2SIMPLE?

• p. 88

Also note financial mathematics problems at end
of TAB Notes on Excel and LOTUS.
3
MINICASE 2

• Why is financial mathematics (time value of
  money) so important in financial analysis?

4

• a.Time lines show timing of cash flows.
• ALWAYS A GOOD IDEA TO DRAW A TIME LINE.

0
1
2
3
i
CF0
CF1
CF3
CF2
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.
5
Time line for a 100 lump sum due at the end of
Year 2.
0
1
2 Years
i
100
6
Time line for an ordinary annuity of 100 for 3
years.
0
1
2
3
i
100
100
100
7
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
3
i
100
50
75
-50
8
b(1) Whats the FV of an initial100 after 3
years if i 10?
0
1
2
3
10
FV ?
100
Finding FVs is compounding.
9
b(1) Whats the FV of an initial100 after 3
years if i 10?
0
1
2
3
10
FV ?
100
110
?
Finding FVs is compounding.
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.
11
After 3 years
FV3 PV(1 i)3 100(1.10)3
133.10.
In general,
FVn PV(1 i)n.
12
Four Ways to Find FVs

• Solve the equation with a regular calculator
• Use tables
• Use a financial calculator
• Use a spreadsheet

13
USING TABLES See handout

• 3 PERIODS
• 10
• 1.3310
• times 100 133.10
• SAY GOOD-BYE TO USING TABLES!

14
Financial Calculator Solution
Financial calculators solve this equation
There are 4 variables. If 3 are known, the
calculator will solve for the 4th.
15
Heres the setup to find FV
3 10 -100 0 N I/YR PV PMT FV
INPUTS
OUTPUT
133.10
Clearing automatically sets everything to 0, but
for safety enter PMT 0.
Set P/YR 1, END
16
b(2) Whats the PV of 100 due in 3 years if i
10?
Finding PVs is discounting, and its the reverse
of compounding.
0
1
2
3
10
100
PV ?
17
Solve FVn PV(1 i )n for PV
PV
FVn (1 i)n
n
PV 100/( ) 100(0.7513) 75.13.
3
1.10
18
Financial Calculator Solution
3 10 0 100 N I/YR PV PMT FV
-75.13
INPUTS
OUTPUT
Either PV or FV must be negative. Here PV
-75.13. Put in 75.13 today, take out 100
after 3 years.
19
EXCEL SOLUTION

• LOOK AT FUNCTIONS PAGE FOR EXCEL/LOTUS.

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

21
Spreadsheet Solution

• Use the PV function see spreadsheet.
• PV(Rate, Nper, Pmt, FV)
• PV(0.10, 3, 0, 100) -75.13

22
c. If sales grow at 20 per year, how long before
sales double?
Solve for n Time line ?
FVn PV(1 i)n 2
1(1.20)n (1.20)n 2 n ln (1.20) ln
2 n(0.1823) 0.6931 n
0.6931/0.1823 3.8 years.
23
20 -1 0 2 N I/YR PV PMT FV
3.8 BewareSome Calculators round up.

INPUTS
OUTPUT
Graphical Illustration
FV
2
3.8
1
Years
0
1
2
3
4
24
Spreadsheet Solution

• Use the NPER function see spreadsheet.
• NPER(Rate, Pmt, PV, FV)
• NPER(0.20, 0, -1, 2) 3.8

Correction
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?

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?

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.

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?

29
d. Whats the difference between an ordinary
annuity and an annuity due?

30
d. Whats the difference between an ordinary
annuity and an annuity due?
Ordinary Annuity
0
1
2
3
i
PMT
PMT
PMT
Annuity Due
0
1
2
3
i
PMT
PMT
PMT
36
31
HINT

• ANNUITY DUE OF n PERIODS IS EQUAL TO A REGULAR
  ANNUITY OF (n-1) PERIODS PLUS THE PMT.

32
e(1). Whats the FV of a 3-year ordinary annuity
of 100 at 10?
0
1
2
3
10
100
100
100
FV
33
e(1). Whats the FV of a 3-year ordinary annuity
of 100 at 10?
0
1
2
3
10
100
100
100
110 121
FV 331
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

35
Financial Calculator Formula for Annuities
Financial calculators solve this equation
There are 5 variables. If 4 are known, the
calculator will solve for the 5th.
Correct but confusing!
36
Financial Calculator Solution
INPUTS
3 10 0 -100 331.00
N
I/YR
PV
PMT
FV
OUTPUT
Have payments but no lump sum PV, so enter 0 for
present value.
37
Spreadsheet Solution

• Use the FV function see spreadsheet.
• FV(Rate, Nper, Pmt, Pv)
• FV(0.10, 3, -100, 0) 331.00

38
e(2). Whats the PV of this ordinary annuity?
0
1
2
3
10
100
100
100
_____
PV
39
Whats the PV of this ordinary annuity?
0
1
2
3
10
100
100
100
90.91
82.64
75.13
248.69 PV
40
3 10 100 0
-248.69
INPUTS
N
PV
PMT
FV
I/YR
OUTPUT
Have payments but no lump sum FV, so enter 0 for
future value.
41
Spreadsheet Solution

• Use the PV function see spreadsheet.
• PV(Rate, Nper, Pmt, Fv)
• PV(0.10, 3, 100, 0) -248.69

42
e(3). Find the FV and PV if theannuity were an
annuity due.
0
1
2
3
10
100
100
100
43
Could, on the 12C, switch from End to Begin
i.e. f Begin. Then enter variables to find PVA3
273.55.
3 10 100 0
-273.55
INPUTS
N
I/YR
PV
PMT
FV
OUTPUT
Then enter PV 0 and press FV to find FV
364.10.
44
Another HINT

• FV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE
  FV OF A REGULAR ANNUITY OF n PERIODS TIMES (1k)
  (slide 30)
• PV OF AN ANNUITY DUE OF n PERIODS IS EQUAL TO THE
  PV OF A REGULAR ANNUITY OF n PERIODS TIMES (1k)

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.

46
PV and FV of Annuity Due vs. Ordinary Annuity

• PV of annuity due
• (PV of ordinary annuity) (1i)
• (248.69) (1 0.10) 273.56
• FV of annuity due
• (FV of ordinary annuity) (1i)
• (331.00) (1 0.10) 364.1

47
Switch from End to Begin. Then enter
variables to find PVA3 273.55.
INPUTS
3 10 100 0
-273.55
N
I/YR
PV
PMT
FV
OUTPUT
Then enter PV 0 and press FV to find FV
364.10.
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)
49
EXCEL SOLUTION
50
(f) What is the PV of this uneven cashflow
stream?
3
0
1
2
4
10
300
100
300
-50
______
PV
51
(f) What is the PV of this uneven cashflow
stream?
3
0
1
2
4
10
300
100
300
-50
90.91
247.93
225.39
-34.15
530.08 PV
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.

53
CALCULATOR SOLUTION

• 0 g CF0
• 100 g CFj
• 300 g CFj
• 2 g Nj
• 50 CHS g CFj
• 10 i
• f NPV

54
EXCEL SOLUTION

• REMEMBER EXCEL/LOTUS READS THE 1ST CASH FLOW AS
  OCCURING ONE PERIOD HENCE.

55
Spreadsheet Solution
A B C D E 1 0 1 2 3 4 2 100 300 300 -50 3 53
0.09
Excel Formula in cell A3 NPV(10,B2E2)
56
g. What interest rate would cause 100 to grow to
125.97 in 3 years?
100 (1 i )3 125.97.
3 -100 0 125.97
INPUTS
N
I/YR
PV
FV
PMT
OUTPUT
8.00
57
EXCEL SOLUTION
58
h.Will the FV of a lump sum be larger or smaller
if we compound more often, holding the stated i
constant? Why?
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.
60
0
1
2
3
10
100
133.10
Annually FV3 100(1.10)3 133.10.
0
1
2
3
0
1
2
3
4
5
6
5
134.01
100
Semiannually FV6 100(1.05)6 134.01.
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.

62

• Effective Annual Rate (EAR EFF)
• The annual rate which causes PV to grow to the
  same FV as under multiperiod compounding.

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.

64
h(3).How do we find EAR for a nominal rate of
10, compounded semiannually?
Or use a financial calculator (not 12C)
65
EAR (1knom/m)m - 1

• .10 ENT
• 2 divide
• 1
• 2 Yx
• 1 -
• .1025

(1 EAR) (1knom/m)m
66
CALCULATOR

• WHAT IS EAR IF COMPOUNDING QUARTERLY?
• COMPOUNDING DAILY?
• COMPOUNDING CONTINUOUSLY?

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.
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

69
i. Can the effective rate ever beequal 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.

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!
71
Used in calculations, shown on time lines.
iPer
If iNom has annual compounding, then iPer
iNom/1 iNom. Can always use EAR!
72
EAR EFF Used to compare returns on
investments with different compounding
patterns. Also used for calculations if dealing
with annuities where payments dont match
interest compounding periods.
73
FV of 100 after 3 years under 10 semiannual
compounding? Quarterly? Daily?
mn
i
?
?
Nom
FV


PV
1 .


?
?
n
?
?
m
2x3
0.10
?
?
FV


100
1


?
?
?
?
3S
2
100(1.05)6 134.01.
FV3Q 100(1.025)12 134.49.
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

75
j(3). Whats the value at the end of Year 3 of
the following CF stream if the quoted interest
rate is 10, compounded semi-annually?
4
5
0
1
2
3
6 6-mos. periods
5
100
100
100
76

• Payments occur annually, but compounding occurs
  each 6 months.
• So we cant use normal annuity valuation
  techniques.

77
1st Method Compound Each CF
0
1
2
3
4
5
6
5
100
100.00
100
110.25
121.55
331.80
FVA3 100(1.05)4 100(1.05)2 100 331.80.
78
Could you find FV with afinancial calculator?
2nd Method Treat as an Annuity I.E. USE EAR
Yes, by following these steps a. Find the EAR
for the quoted rate
EAR (1 ) - 1 10.25.
2
0.10 2
79
Or, to find EAR with a 17 OR 19b Calculator
NOM 10 P/YR 2 EFF 10.25
80
b. The cash flow stream is an annual
annuity whose EFF 10.25.
c.
3 10.25 0 -100
INPUTS
N
I/YR
PV
FV
PMT
OUTPUT
331.80
81
j(2) Whats the PV of this stream?
0
1
2
3
5
100
100
100

82
Whats the PV of this stream?
0
1
2
3
5
100
100
100
90.70 82.27 74.62 247.59
83
Calculator solution

• 100 PMT
• 3 n
• 10.25 i
• f NPV
• 247.59

84
Whats the FV of this stream under quarterly
compouning?
0
1
2
3
100
100
100

85
EAR WORKSHEET
86
k. Amortization
Construct an amortization schedule for a 1,000,
10 annual rate loan with 3 equal payments.
87
Step 1 Find the required payments.
0
1
2
3
10
PMT
PMT
PMT
-1000
3 10 -1000
0
INPUTS
N
I/YR
PV
FV
PMT
OUTPUT
402.11
88
ALGEBRA

• PMT/(1k) PMT/(1k)2 PMT/(1k)3 1000
• PMT 1/(1k) 1/(1k)2 1/(1k)3 1000, or
• PMT
  1000/ 1/(1k) 1/(1k)2 1/(1k)3

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.
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.
91
BEG PRIN END YR BAL PMT INT REDUCTION BAL
1 1,000 402 100 302 698 2 698 402 70 332 36
6 3 366 402 37 366 0 TOT 1,206.34 206.34 1,000
Interest declines. Tax implications.
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.
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.

94
EXCEL SOLUTION
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.)
96
iPer 10.0 / 365 0.027397 per day.
0
1
2
273
0.027397
FV?
-100
273
?
?
FV


100
1.00027397
273
?
?


100
1.07765


107.77.
Note in calculator, decimal in equation.
97
iPer iNom/m 10.0/365 0.027397 per day.
INPUTS
273 -100 0
107.77
I/YR
PV
FV
N
PMT
OUTPUT
Enter i in one step. Leave data in calculator.
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.
99
iPer 0.027397 per day.
0
365
638 days
-100
FV 119.10
FV 100(1 0.10/365)638 100(1.00027397)638
100(1.1910) 119.10.
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

101

• MORE PRECISELY, instead of .75 exponent
• Calculate 1EAR as above, then
• 273 ENTER
• 365 DIVIDE (1EAR)
• Yx (1EAR)(273/365)
• 100 MULTIPLY
• 107.77

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?

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

104
m(1 EAR)(1/m) - 1 knom Using the
calculator, EAR 10, dailycompounding.

• 1.1 ENTER
• 365 1/X Yx
• 1 -
• 365 MULTIPLY
• 9.53

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 dont buy the note.
The note is riskless. Should you buy it?
106
iPer 0.019178 per day.
0
365
456 days
1,000
-850
3 Ways to Solve 1. Greatest future wealth
FV 2. Greatest wealth today PV 3. Highest
rate of return Highest EFF
107
1. Greatest Future Wealth
Find FV of 850 left in bank for 15 months and
compare with notes FV 1000.
FVBank 850(1.00019178)456 927.67 in bank.
Buy the note 1000 927.67.
108
Calculator Solution to FV
iPer iNom/m 7.0/365 0.019178 per day.
INPUTS
456 -850 0
927.67
N
I/YR
PV
FV
PMT
OUTPUT
Enter iPer in one step.
109
2. Greatest Present Wealth
Find PV of note, and compare with its 850 cost
PV 1000/(1.00019178)456 916.27.
110
7/365
INPUTS
456 .019178 0
1000
-916.27
N
I/YR
PV
FV
PMT
OUTPUT
PV of note is greater than its 850 cost, so buy
the note. Raises your wealth.
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.
112
456 -850 0 1000
0.035646 per day

INPUTS
N
I/YR
PV
FV
PMT
OUTPUT
Convert to decimal
Decimal 0.035646/100 0.00035646.
EAR EFF (1.00035646)365 - 1
13.89.
113
Using interest conversion P/YR 365 NOM 0
.035646(365) 13.01 EFF 13.89 Since 13.89
7.25 opportunity cost, buy the note.
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?

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?

116
JOHNM PROBLEMS