1. CHAPTER 2 Discounted Cash Flow Analysis Time Value of Money Financial Mathematics
Future value
Present value
Rates of return
Amortization
Annuities, AND
Many Examples
Some of later examples change the interest rate from 10% to 12%. I did not do this, since, I then would have to change the excel spreadsheet.

2. MINICASE 2 SIMPLE?
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. 1 2 3 i%
CF0
CF1
CF2
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.

5. Time line for a $100 lump sum due at the end of Year 2.1
2 Years i%
100

6. Time line for an ordinary annuity of $100 for 3 years.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 3.1 2 3 i%
-50
100 75 50

8. b(1) What’s the FV of an initial $100 after 3 years if i = 10%?1 2 3
10%
100
FV = ?
Finding FVs is compounding.

9. b(1) What’s the FV of an initial $100 after 3 years if i = 10%?1 2 3
10%
100
FV = ?
110 ?
Finding FVs is compounding.

10. After 1 year
FV1 = PV + INT1 = PV + PV(i)
= PV(1 + i)
= $100(1.10)
= $
After 2 years
FV2 = FV1(1 + i)
= PV(1 + i)2
= $100(1.10)2
= $

11. After 3 years
FV3 = PV(1 + i)3
= 100(1.10)3
= $
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 % =
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. Here’s the setup to find FV:
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) What’s the PV of $100 due in 3 years if i = 10%?
Finding PVs is discounting, and it’s the reverse of compounding. 1 2 3
10%
PV = ?
100

17. Solve FVn = PV(1 + i )n for PV:
= $100(0.7513) = $75.13. 3
1.10

18. Financial Calculator Solution
N I/YR PV PMT FV
-75.13
INPUTS
OUTPUT
Either PV or FV must be negative. Here
PV = Put in $75.13 today, take
out $100 after 3 years.

19. EXCEL SOLUTION
LOOK AT FUNCTION’S 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) =

21. Spreadsheet Solution
Use the PV function: see spreadsheet.
= PV(Rate, Nper, Pmt, FV)
= PV(0.10, 3, 0, 100) =

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) =
n = / = 3.8 years.

23. Graphical Illustration:
N I/YR PV PMT FV
3.8 Beware:Some Calculators round up.
INPUTS
OUTPUT
Graphical Illustration: FV 2
3.8 1
Years 1 2 3 4

24. Use the NPER function: see spreadsheet. = NPER(Rate, Pmt, PV, FV)
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. n * krequired to double = 72 In this case,
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. What’s the difference between an ordinary annuity and an annuity due?

30. d. What’s the difference between an ordinary annuity and an annuity due?1 2 3 i%
PMT
PMT
PMT
Annuity Due 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). What’s the FV of a 3-year ordinary annuity of $100 at 10%?1 2 3
10%
100
100
100
FV =

33. e(1). What’s the FV of a 3-year ordinary annuity of $100 at 10%?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
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) =

38. e(2). What’s the PV of this ordinary annuity?1 2 3
10%
100
100
100
_____
= PV

39. What’s the PV of this ordinary annuity?1 2 3
10%
100
100
100
90.91
82.64
75.13
= PV

40. Have payments but no lump sum FV, so enter 0 for future value.
INPUTS N
I/YR PV
PMT FV
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) =

42. e(3). Find the FV and PV if the annuity were an annuity due.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 = $
INPUTS N
I/YR PV
PMT FV
OUTPUT
Then enter PV = 0 and press FV to find
FV = $

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

45. HINT, illlustrated
The PV of this regular annuity was
Multiply this by ( ), and you get: , 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) (1+i)
= (248.69) ( ) =
FV of annuity due:
= (FV of ordinary annuity) (1+i)
= (331.00) ( ) = 364.1

47. Switch from “End” to “Begin”.
Then enter variables to find PVA3 = $
INPUTS N
I/YR PV
PMT FV
OUTPUT
Then enter PV = 0 and press FV to find
FV = $

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 cash flow stream?1 2 3 4
10%
100
300
300
-50
______
= PV

51. (f) What is the PV of this uneven cash flow stream?1 2 3 4
10%
100
300
300
-50
90.91
247.93
225.39
-34.15
= PV

52. Input in “CFLO” register:
Enter I = 10%, then press NPV button to get NPV =

53. CALCULATOR SOLUTION
g CF0
100 g CFj
300 g CFj
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 3 530.09
Excel Formula in cell A3:
=NPV(10%,B2:E2)

56. g. What interest rate would cause $100 to grow to $125.97 in 3 years?
INPUTS N
I/YR PV
PMT FV
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. 1 2 3
10%
100
133.10
Annually: FV3 = 100(1.10)3 = 1 2 3 1 2 3 4 5 6 5%
100
134.01
Semiannually: FV6 = 100(1.05)6 =

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 = %.

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 = (1+knom/m)m - 1
ENT
divide Yx =
.1025
(1 + EAR) = (1+knom/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 = ( /12) = %.
EARD(360) = ( /360) = %.

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 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.

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. Can always use EAR! iPer: Used in calculations, shown on time lines.
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
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.

73. FV of $100 after 3 years under 10% semiannual compounding. Quarterly
FV of $100 after 3 years under 10% semiannual compounding? Quarterly? Daily? mn i   FV = PV  +
Nom  n   m
2x3
0.10   FV =
$100  1 +  3S   2
= $100(1.05)6 = $
FV3Q = $100(1.025)12 = $

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). 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? 1 2 3 4 5
6 6-mos.
periods 5%
100
100
100

76. Payments occur annually, but compounding occurs each 6 months.
So we can’t use normal annuity valuation techniques.

77. 1st Method: Compound Each CF1 2 3 4 5 6 5%
100
100
100.00
110.25
121.55
331.80
FVA3 = 100(1.05) (1.05) =

78. Could you find FV with a financial calculator?
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 = ( ) - 1 = 10.25%.
0.10 2 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%.
INPUTS N
I/YR PV
PMT FV
OUTPUT
331.80

81. j(2) What’s the PV of this stream?1 2 3 5%
100
100
100

82. What’s the PV of this stream?1 2 3 5%
100
100
100
90.70
82.27
74.62
247.59

83. Calculator solution
PMT n i
f NPV =
247.59

84. What’s the FV of this stream under quarterly compouning?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.1 2 3
10%
-1000
PMT
PMT
PMT
INPUTS N
I/YR PV
PMT FV
OUTPUT
402.11

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 = / [1/(1+k) + 1/(1+k)2 + 1/(1+k)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 =
= $

90. Step 4: Find ending balance after Year 1.
End bal = Beg bal - Repmt
= 1, = $
Repeat these steps for Years 2 and 3
to complete the amortization table.

91. Interest declines. Tax implications.
BEG PRIN END
YR BAL PMT INT REDUCTION BAL
1 $1,000 $402 $100 $302 $698
TOT 1, ,000
Interest declines. Tax implications.

92. 10% on loan outstanding, which is falling.$
402.11
Interest
302.11
Principal Payments 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. FV = $100 1.00027397 =1 2
273 %
-100
FV=?  
273 FV =
$100
273 =
$100   = $
Note: % in calculator, decimal in equation.

97. Leave data in calculator.
iPer = iNom/m
= 10.0/365
= % per day.
INPUTS
107.77 N
I/YR PV
PMT FV
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
Now suppose you leave your money in the bank for 21 months, which is 1.75 years or = 638 days.
How much will be in your account at maturity?
Answer: Override N = 273 with N = FV = $

99. iPer = 0.027397% per day. FV = $100(1 + 0.10/365)638
365
638 days
-100
FV =
FV = $100( /365)638
= $100( )638
= $100(1.1910)
= $

100. ALTERNATIVE SOLUTION USING EAR
FIND EAR
ENTER
DIVIDE
Yx [= (1 + EAR)]
Yx [= (1 + EAR).75]
MULTIPLY
= $107.79

101. MORE PRECISELY, instead of .75 exponent:
Calculate [1+EAR] as above, then
ENTER
DIVIDE [=(1+EAR)]
Yx [=(1+EAR)(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%, daily compounding.
ENTER
/X Yx
365 MULTIPLY
= %

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 % 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?

106. 1. Greatest future wealth: FV 2. Greatest wealth today: PV
iPer = % per day.
365
456 days
-850
1,000
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
note’s FV = $1000.
FVBank = $850( )456
= $ in bank.
Buy the note: $1000 > $

108. Calculator Solution to FV:
iPer = iNom/m
= 7.0/365
= % per day.
INPUTS
927.67 N
I/YR PV
PMT FV
OUTPUT
Enter iPer in one step.

109. 2. Greatest Present Wealth
Find PV of note, and compare
with its $850 cost:
PV = $1000/[( )456]
= $

110. 7/365 =
INPUTS N
I/YR PV
PMT FV
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. Convert % to decimal: Decimal = 0.035646/100 = 0.00035646.
INPUTS %
per day N
I/YR PV
PMT FV
OUTPUT
Convert % to decimal:
Decimal = /100 =
EAR = EFF% = ( )
= 13.89%.

113. Using interest conversion:
P/YR = 365
NOM% = (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, 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, WITH A NOMINAL DISCOUNT RATE OF 10 PERCENT, QUARTERLY COMPOUNDING, WHAT IS THE PV OF THE NOTE?

116. JOHNM PROBLEMS