1. FIN 360: Corporate Finance
Topic 5: Time Value of Money II:
Cash Flow Streams
Larry Schrenk, Instructor

2. Today’s Outline General Notes Time Value of Money Rates of Change
Mixed Cash Flows
Annuities
Perpetuities
Non-Annual Cash Flows
Rates of Change
Amortization

3. 1. General Notes

4. Time Lines The Use of Time Lines Temporal Indices: t T 1, 2, 3,… 1 2 31 2 3 4 5 6 T C0 C1 C2 C3 C4 C5 C6 CT

5. Negatives Possible Representations Calculation: Same as Positive -10
(10) 10
Note: I will never use just color indicate a negative value.
Calculation: Same as Positive

6. ‘Negatives’ Rule For problems covered in the this topic.
Of the dollar values entered from the data
Make one and only one negative.
It does not matter which one.

7. 2. Time Value of Money A. Mixed Cash Flows

8. Mixed Cash Flows Non-Constant Examples
Calculate individually, them sum. 1 2 3 4
$101.24
$200.12
-$300.34
$2.23 1 2 3 4
$55.00
-$16.00

9. Mixed Cash Flows: Example1 2 3 4
$55.00
-$16.00
Assume a 10% discount rate
N = 1; I/Y = 10; FV = -55; PV = 50.00
N = 2; I/Y = 10; FV = -(-16); PV =
N = 3; I/Y = 10; FV = -55; PV = 41.32
N = 4; I/Y = 10; FV = -(-16); PV =
Sum: – – = $67.20
Note: Easier Calculator Function Later.

10. B. Annuities

11. Annuities Annuities A finite series of constant cash flows, e.g.,
$100 per year for 5 years
$10 per month for 7 months
Variables
Cash Flow Amount (per period)
PMT key on your Calculator
Date of the First Payment
Period (weekly, quarterly, annually)
Length (or Number) of Payments

12. Annuities A 5 year annual, annuity of $50 beginning in year 1:
TECHNICAL NOTE: When we use the term ‘annuity’ we mean an ‘annuity in arrears’, i.e., an annuity whose first payment begins next (not this) period. An annuity that begins this period is called an ‘annuity due’ and will be considered toward the end of the lecture. 1 2 3 4 5 6 50 50 50 50 50

13. Payments begin next period.
Annuity Time Line
The constant cash flows of a 3 year annuity of $ per year at 10%:
I/Y
I/Y
I/Y 1 2 3
PMT
PMT
PMT
10%
10%
10% 1 2 3
Payments begin next period.
$100.00
$100.00
$ ▪

14. Annuities
Annuities can always be valued as a series of one time cash flow (r = 7%): 1 2 3 4 5 6 50 50 50 50 50

15. Uses of Annuity: PV Present Value of an Annuity
‘Borrowing’ Problems (Loans)
How much can I borrow (PV)?
How much are my payments (PMT)?
What interest rate am I getting (I/Y)?
How long will it take (N)?
‘Value Now’ Problems
How much is it worth now(PV)?
How much are my cash flows(PMT)?
What is the interest rate (I/Y)?
How long does it go on(N)?

16. Uses of Annuity: FV Future Value of an Annuity
‘Savings’ Problems (Retirement Fund)
How much will I have (FV)?
How much are my payments (PMT)?
What interest rate am I getting (I/Y)?
How long will it take (N)?

17. Annuities: PV Formula Present Value Formula:
NOTE: The formula assumes that the first payment begins next period!

18. Annuities: PV Example Thus:
N = 5; I/Y = 7; PV = $205.01; PMT = -50; FV = 0 1 2 3 4 5 6 50 50 50 50 50

19. Annuities: FV Formula Future Value Formula :
NOTE: The formula assumes that the first payment begins next period and it give the future value in year T!

20. Annuities: FV Example Thus:
N = 5; I/Y = 7; PV = 0; PMT = -50; FV = $287.54 1 2 3 4 5 6 50 50 50 50 50

21. Cash Flow Problems

22. Cash Flow, Time and Interest Rates
Annuities have three additional variables:
Cash Flow
Interest Rate
Time
PV and FV Versions

23. Cash Flow Problems The are Two Versions of Cash Flow Problems
Present Value: What it the payment on a loan?
Future Value: What do you need to save to achieve an investment goal?

24. PV Cash Flow Problems Payments on a loan.
For example, if I borrow $1, at 11% for 5 years, what are my payments?
That is a question of calculating the cash flows of an annuity.
N = 5; I/Y = 11; PV = -1,000; PMT = ; FV = 0

25. FV Cash Flow Problems
Future value cash flow problems are the ‘savings’ problems.
For example, if I want to have $100, in 5 years and the rate of interest is 11%, how much do I have to save per year?
N = 5; I/Y = 11; PV = 0; PMT = 16,057.03; FV = -100,000

26. Interest Rate Problems

27. Interest Rate Problems
The are Two Versions of Interest Rate Problems
Present Value: What it the interest rate on a loan?
Future Value: What interest rate do you need to achieve an investment goal?

28. PV Interest Rate Problems
Interest rate on a loan
For example, if I borrow $1, for 5 years and repay it $250 per year, what is the interest rate?
That is a question of calculating the interest on a loan.
N = 5; I/Y = 7.93%; PV = 1,000; PMT = -250; FV = 0

29. FV Interest Rate Problems
Interest rate on investments
For example, if I want to have $1, in 5 years and can save $150 per year, what is the interest rate?
That is a question of calculating the interest on an investment.
N = 5; I/Y = 14.43%; PV = 0; PMT = -150; FV = 1,000

30. Time Problems

31. Time Problems The are Two Versions of Time Problems
Present Value: How long will it take to repay a loan?
Future Value: How long will it take to save a certain amount?

32. PV Time Problems
If I borrow $1, at 11%, and I want to make annual payments of $350.00, how long will it take me to repay the loan?
N = 3.62; I/Y =11; PV = -1,000; PMT = 350; FV = 0
3.62 =
0.62 x 12 = 7.44  7
3 years 7 months

33. FV Time Problems
I need $1, and can save $150 per year. If I can get a return of 11%, how long will it take me to reach my goal?
N = 5.27; I/Y =11; PV = 0; PMT = 150; FV = -1,000
5.27 =
0.27 x 12 = 3.24  3
5 years 3 months

34. Annuities Due

35. Annuities Due Cash Flow Begins Now Method Not Next Period
Calculator: Change ‘END’ to ‘BEGIN’

36. Annuities Due
An annuity due is an annuity that begins this period, not next.
Five Year Annuity of $100 per Year
Five Year Due Annuity of $100 per Year 1 2 3 4 5 6 50 50 50 50 50 1 2 3 4 5 6 50 50 50 50 50

37. Annuities Due
What is the value of a 5 year annual, annuity due of $500 (r = 5%)?
BEG/END = BEG; N = 5; I/Y = 5; PV = $2,272.98; PMT = -500; FV = 0

38. C. Perpetuities

39. Perpetuities Perpetuity
An infinite series of constant cash flows, e.g.,
$100 per year forever
$10 per month forever
Variables
Cash Flow Amount (per period)
Date of the First Payment
Period (weekly, quarterly, annually)

40. Perpetuities Valuing a Perpetuity Note:
Since perpetuities are infinite, they cannot have a future value.

41. Perpetuity Example EXAMPLE
What is the present value of $1,000 per year (r = 10%)

42. Growing Perpetuities A Growing Perpetuity
An infinite series of changing cash flows, e.g.,
If g = 5%, then
$100.00, $105.00, , , …
Variables
Cash Flow Amount
The Date of the First Payment
The Period (weekly, quarterly, annually)
Growth Rate

43. Growing Perpetuities Valuing a Growing Perpetuity Three Notes:
The growth can be positive or negative, e.g., the cash flow can either increase or decline at x% per period.
C1 refers to next period’s cash flow.
There are economic reasons why g is never greater than r.

44. Growing Perpetuities EXAMPLE
What is the present value of $1,000 per year growing at 3% per year (r = 10%)

45. Growing Perpetuities EXAMPLE
What is the present value of $1,000 per year declining at 3% per year (r = 10%)

46. Cash Flow and Interest Rate Problems
It is easy to solve perpetuity formula
For the cash flow and
The interest rate.

47. Cash Flow and Interest Rate Problems
A perpetuity is valued at $5,000 and the interest rate is 7%. What is its cash flow?
A perpetuity valued at $5,000 has a cash flow is $400. Find the rate?

48. D. Non-Annual Cash Flows

49. Non-Annual Periods m = Periods per Year Examples 2 = Semi-Annual
4 = Quarterly
12 = Monthly
52 = Weekly
364 or 360 = Daily

50. Calculator Adjustments
Two Changes:
Periods per Year (P/Y)
Adjust P/Y to m
Weekly P/Y = 52
Number of Periods (N)
Remember N is the Number of Periods
Monthly Discounting for 5 Years
N = 12 x 5 = 60

51. Changing P/Y TI
[2nd ] [I/Y]
m [Enter] HP m
[Orange]
PMT

52. FV and Compounding Period

53. PV and Compounding Period

54. Single Dollar
How much do we have after 2 years if we deposit $500 and the interest rate is 10% (compounded quarterly)?
Set P/Y = 4
Input 8, Press N (2 x 4 = 8)
Input 10, Press I/Y (annual rate)
Input 500, press +/-, press PV (you get -500)
Press CPT, FV to get $609.20

55. Non-Annual Annuities
Unfortunately, not all annuities have annual cash flows:
Bonds Semi-annual coupons,
Loans Monthly payments,
Dividends Quarterly,
We can put money in a bank quarterly, weekly, daily or even hourly.
Need a mechanism for adapting all of our annuity formulae for non-annual periods.

56. Annuity FV with a Calculator
How much do we have after 3 years if we save $200 per month beginning next month and the interest rate is 12%?
Set P/Y = 12
Input 0, Press PV
Input 36, Press N (3 x 12 = 36)
Input 12, Press I/Y
Input 200, press +/-, press PMT (you get -200)
Press CPT, FV to get $8,615.38

57. Annuity PV with a Calculator
You have a loan of $10,000 to be repaid in monthly installments over 5 years with an interest rate of 15%? What is the monthly payment?
Set P/Y = 12
Input 0, Press FV
Input 60, Press N (5 x 12 = 60)
Input 15, Press I/Y
Input 10,000, press +/-, press PV (you get -10,000)
Press CPT, PMT to get $237.90

58. Non-Annual Practice Problems
How much will you have if you save $ per month for 25 years at 8%?
$95,102.64
How much can you borrow if you pay $50.00 per week for 5 years at 7%?
$10,962.57
How much do you need to save per month to have $10,000 in 5 years at 10%? ▪
$ ▪

59. Non-Annual Perpetuities
Formula:
Remember that C is the period cash flow.

60. T-S-P
If you increase the number of periods per year, the present value will:
Increase
Remain the same
Decrease
Cannot determine

61. 3. Rates of Change

62. Percentages
Using Absolute (Dollar) Value versus Ratios (e.g., Percentages)
Numerical Representation of Percentages
Integer Form 5%
Decimal Form 0.05
If in doubt, use the decimal form!

63. Percentages Calculating a Percentage Basis Points
If you have 35 balls and 12 are red, what is the percentage of red balls?
Basis Points
A ‘basis point’ is 1/100 of a percentage
1% = 100 basis points
0.25% = 25 basis points

64. Types of Rate of Change Problem
Three types of change are central:
Returns: Change of Dollar Value over Time
Growth Rates: Change of Size over Time
Inflation: Change of Prices over Time

65. Simple Rates (Interest)
Returns
Return on your principal, but
No return on the accumulated interest
$100 in an account for three year at 12% simple interest
= $136.

66. Compound Rates (Interest)
Returns
Return on your principal, and
Return on the accumulated interest
$100 in an account for three year at 12% compound interest
A gain of $4.49 over simple interest!
100.00 1
100.00*1.12
112.00 2
112.00*1.12
125.44 3
125.44*1.12
140.49

67. Holding Period Return Most basic rate calculation
Change from one point of time (t = 0) to another (t = 1):

68. Holding Period Return
My portfolio was worth $123,000 5 years ago and it is now worth $131,000:
REMEMBER: The earlier value always goes in the denominator!

69. Holding Period Return
Problem: Comparing assets with different holding periods.
Which is better?
7.8% over 7 years
10.5% over 10 year
Need a common time period
Convert all rates to an annual basis
‘Annualize’ them (as with ratios)

70. Non-Annual Rates For example, monthly data for stock returns.
If a stock was at $110 at the end of last month and $108 at the end of this month:
Need to annualize the return.

71. Rate Conversions

72. Rate Conversions
Most often we will be converting a non-annual rate to an annual rate.
Unfortunately, there are several ‘versions’ of annual rates.

73. Conversions
HPR
EAR
APR

74. Annual Percentage Rate (APR)
This is an application of simple (not compound) interest.
AKA: Nominal, Stated, Quoted Rate

75. APR Example If you have a monthly HPR of 2%
But if I put $100 in an account at 2% per month and left it there for 12 months, I would have:
So the APR understates my return by 2.82%!

76. Effective Annual Return (EAR)
The correct annual rate to use is the Effective Annual Return (EAR).
This form of the annual rate recognizes compound interest.
AKA:
Equivalent Annual Return (EAR)

77. Effective Annual Return (EAR)
If you have an APR and want to convert it to EAR:
In our example, we had an APR of 24%.

78. Effective Annual Return (EAR)
If you have an HPR and want to convert it to EAR:
In our example, we had a monthly rate of 2%.

79. IMPORTANT DISTINCTION
Formula: APR  EAR
Formula: HPR  EAR

80. Effective Annual Return (EAR)
We can also start with the EAR and find any equivalent HPR.
If my EAR is 31%, then the equivalent weekly HPR is:

81. Calculator Functions Nom = Nominal Rate (APR)
Eff = Effective Rate (EAR)

82. Rate Practice HPRweekly = 0.2%. Find EAR and APR.
APRweekly = 20%. Find EAR
EAR = 10%. Find HPRquarterly. ▪

83. 4. Amortization

84. Amortization Installment Loan Compound Interest Equal Payments
Distinguish
Repayment of Principle
Repayment of Interest

85. Amortization Graph

86. Amortization Example Loan: $1,000 Maturity: 3 years Interest Rate: 8%
Period: Annual
Calculate Equal Payments
N =3; I/Y = 8; PV = -1,000; PMT = $388.03; FV = 0

87. Amortization Calculation
Find Interest due on Balance.
Subtract Interest from Payment to get Principle.
Subtract Principle from Balance to get New Balance.

88. Amortization Calculation
Year 1 2 3
Begin Bal.
1,000
691.97
359.29
Payment
388.03
Interest
80.00
55.36
28.74
Interest
80.00
55.36
28.74
Principal
308.03
332.68
359.29
Principal
308.03
332.68
359.29
End Bal.
691.97
359.29
End Bal.
691.97
359.29
Interest = Rate x Balance
359.29(0.08) = 28.74
691.97(0.08) = 55.36
1,000(0.08) = 80
Principal = Payment – Interest
– 80 =
– =
– =
Balance = Balance – Principal
– =
1,000 – =
– = 0