1. CHAPTER 7 Time Value of Money
Future value
Present value
Rates of return
Amortization

2. Time lines show timing of cash flows.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.

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

4. Time line for an ordinary annuity of $100 for 3 years.1 2 3 i%
100
100
100

5. 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

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

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

8. After 3 years:
FV3 = PV(1 + i)3
= 100(1.10)3
= $
In general,
FVn = PV(1 + i)n.

9. Four Ways to Find FVs
Solve the equation with a regular calculator.
Use tables.
Use a financial calculator.
Use a spreadsheet.

10. Financial Calculator Solution
Financial calculators solve this equation:
FVn = PV(1 + i)n.
There are 4 variables. If 3 are known, the calculator will solve for the 4th.

11. Here’s the setup to find FV:
INPUTS
N I/YR PV PMT FV
133.10
OUTPUT
Clearing automatically sets everything to 0, but for safety enter PMT = 0.
Set: P/YR = 1, END

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. 1 2 3
10%
PV = ?
100

13. ( ) ( ) Solve FVn = PV(1 + i )n for PV: . 1 æ ö PV = $100 ç ÷ = $1003 1 æ ö ( ) PV =
$100 ç ÷ =
$100
PVIF è ø i, n
1.10 =
$100 (
0.7513 ) =
$75.13.

14. 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.

15. If sales grow at 20% per year, how long before sales double?
Solve for n:
FVn = 1(1 + i)n;
2 = 1(1.20)n
Use calculator to solve, see next slide.

16. Graphical Illustration:
INPUTS
N I/YR PV PMT FV
3.8
OUTPUT
Graphical Illustration: FV 2
3.8 1
Year 1 2 3 4

17. 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

18. 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

19. 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.

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

21. 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.

22. Find the FV and PV if the annuity were an annuity due.1 2 3
10%
100
100
100

23. 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 = $

24. 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

25. Input in “CFLO” register:
Enter I = 10, then press NPV button to get NPV = (Here NPV = PV.)

26. What interest rate would cause $100 to grow to $125.97 in 3 years?
INPUTS N
I/YR PV
PMT FV
OUTPUT 8%

27. 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.

28. 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 =

29. We will deal with 3 different rates:
iNom = nominal, or stated, or
quoted, rate per year.
iPer = periodic rate.
EAR = EFF% =
effective annual
rate

30. iNom is stated in contracts. Periods per year (m) must also be given.
Examples:
8%; Quarterly
8%, Daily interest (365 days)

31. 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 = %.

32. Effective Annual Rate (EAR = EFF%):
The annual rate that causes PV to grow to the same FV as under multi-period compounding.
Example: EFF% for 10%, semiannual:
FV = (1 + iNom/m)m
= (1.05)2 =
EFF% = 10.25% because
(1.1025)1 =
Any PV would grow to same FV at 10.25% annually or 10% semiannually.

33. 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.

34. How do we find EFF% for a nominal rate of 10%, compounded semiannually?
Or use a financial calculator.

35. EAR = EFF% of 10% EARAnnual = 10%. EARQ = (1 + 0.10/4)4 – 1 = 10.38%.
EARM = ( /12)12 – 1 = %.
EARD(360) = ( /360)360 – 1 = %.

36. 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.

37. When is each rate used?
iNom:
Written into contracts, quoted by banks and brokers. Not used in calculations or shown
on time lines.

38. iPer:
Used in calculations, shown on time lines.
If iNom has annual compounding,
then iPer = iNom/1 = iNom.

39. EAR = EFF%:
Used to compare returns on investments with different payments per year.
(Used for calculations if and only if
dealing with annuities where payments don’t match interest compounding periods.)

40. FV of $100 after 3 years under 10% semiannual compounding? Quarterly?mn i æ ö FV = PV ç +
Nom ÷ n è ø m
2x3
0.10 æ ö FV =
$100 ç 1 + ÷ 3S è ø 2
= $100(1.05)6 = $
FV3Q = $100(1.025)12 = $

41. What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 1 2 3 4 5
6 6-mos.
periods 5%
100
100
100

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

43. 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) =

44. Could you find FV with a financial calculator?
2nd Method: Treat as an Annuity
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

45. Or, to find EAR with a calculator:
NOM% = 10.
P/YR = 2.
EFF% =

46. b. The cash flow stream is an annual annuity. Find kNom (annual) whose
EFF% = 10.25%. In calculator,
EFF% = 10.25
P/YR = 1
NOM% = 10.25 c.
INPUTS N
I/YR PV
PMT FV
OUTPUT
331.80

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

48. Amortization
Construct an amortization schedule
for a $1,000, 10% annual rate loan
with 3 equal payments.

49. Step 1: Find the required payments.1 2 3
10%
-1,000
PMT
PMT
PMT
INPUTS N
I/YR PV
PMT FV
OUTPUT
402.11

50. 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
= $ – $100
= $

51. Step 4: Find ending balance after Year 1.
End bal = Beg bal – Repmt
= $1,000 – $ = $
Repeat these steps for Years 2 and 3
to complete the amortization table.

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

53. 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.

54. 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.

55. 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.)

56. ( ) ( ) iPer = 10.0% / 365 = 0.027397% per day. ... FV = $1001 2
273 %
...
-100
FV = ? FV =
$100 ( )
273
273 =
$100 ( ) = $
Note: % in calculator, decimal in equation.

57. 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.

58. 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 = 638.
FV = $

59. iPer = 0.027397% per day. ... ... FV = $100(1 + .10/365)638
365
638 days
...
...
-100
FV =
FV = $100( /365)638
= $100( )638
= $100(1.1910)
= $

60. You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that 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?

61. 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%

62. 1. Greatest Future Wealth
Find FV of $850 left in bank for
15 months and compare with
note’s FV = $1,000.
FVBank = $850( )456
= $ in bank.
Buy the note: $1,000 > $

63. 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.

64. 2. Greatest Present Wealth
Find PV of note, and compare
with its $850 cost:
PV = $1,000/( )456
= $

65. 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.

66. 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
$1,000 = $850(1 + i)456
Now we must solve for i.

67. 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% = ( )365 – 1
= 13.89%.

68. Using interest conversion:
P/YR = 365.
NOM% = (365) =
EFF% =
Since 13.89% > 7.25% opportunity cost,
buy the note.