1. NPV and the Time Value of Money
Chapter 4
NPV and the
Time Value of Money

2. Chapter Outline 4.1 The Timeline 4.2 Solving “Single Sum” problems
4.3 Valuing a Stream of Cash Flows
4.4 The Net Present Value of a Stream of Cash Flows
4.5 Annuities and Perpetuities
5.2 Discount Rates and Loans
Amortization Schedule

3. Learning Objectives How to construct a cash flow timeline
Calculate Present Value and Future Value of a Single Sum
Value a series of cash flows and compute the net present value (NPV)
Uneven Cash Flows
Calculate Present Value and Future Value of multiple cash flows:
Annuities and Perpetuities
Calculate loan payments and construct loan amortization schedules

4. Types of Time Value Problems
Single Sum: One dollar amount
Uneven Cash Flows: A series of unequal cash flows
Net Present Value (NPV)
Annuity: A series of equal cash flows for a specified number of periods at a constant interest rate
Perpetuity: An infinite series of equal payments

5. Basic Terms Present Value (PV)– Value as of today
Future Value (FV) – Value at some point in the future
Interest rate (r) – rate of return (FV) or discount rate (PV)
Cost of capital
Required return
Number of Periods (n) - usually measured in months or years
Payment (C) – used for annuity problems
It’s important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an “interest rate” only in terms of loans and savings accounts. The issue of understanding terminology is laid out in more detail in a Lecture Tip in the IM.

6. 4.1 The Timeline
A series of cash flows lasting several periods is defined as a stream of cash flows. We can represent a stream of cash flows on a timeline, a linear representation of the timing of the expected cash flows.

7. Constructing a Timeline
Date 0 represents the present. Date 1 is one year later

8. Distinguishing Cash Inflows from Outflows
The first cash flow at date 0 (today) is represented as negative because it is an outflow

9. Example 4.1 Constructing a Timeline
Suppose you must pay tuition of $10,000 per year for the next four years. Your tuition payments must be made in equal installments of $5,000 each every 6 months. What is the timeline of your tuition payments?

10. Example 4.1 Constructing a Timeline
Assuming today is the start of the first semester, your first payment occurs at date 0 (today). The remaining payments occur at 6-month intervals. Using one-half year (6 months) as the period length, we can construct a timeline as follows:

11. 4.2 Valuing Cash Flows at Different Points in Time
Three Important Rules Central to Financial Decision Making
Rule 1: Comparing and Combining Values
Rule 2: Compounding – Future Value
Rule 3: Discounting – Present Value

12. Rule 1: Comparing and Combining Values
A dollar today and a dollar in one year are not equivalent. Having money now is more valuable than having money in the future; if you have the money today you can earn interest on it.

13. Rule 2: Compounding
The process of moving forward along the timeline to determine a cash flow’s value in the future (its future value) is known as compounding.

14. Rule 2: Compounding
We can apply this rule repeatedly. Suppose we want to know how much the $1,000 is worth in two years’ time. If the interest rate for year 2 is also 10%, then:

15. Rule 2: Compounding
This effect of earning interest on both the original principal plus the accumulated interest, so that you are earning “interest on interest,” is known as compound interest.
Compounding the cash flow a third time, assuming the competitive market interest rate is fixed at 10%, we get:

16. Figure 4.1: The Compounding of Interest over Time

17. Future Value of a Single Sum: Text Book Formula
(Eq. 4.1)

18. Rule 3: Discounting
The process of finding the equivalent value today of a future cash flow is known as discounting.  

19. Present Value of a Single Sum Text Book Formula
(Eq. 4.2)

20. Single Sum Formulas Simplified
No need to have “C” in both formulas. So we will replace it to simplify the equation and, hopefully, avoid using the wrong formula:
Future Value Formula:
FV = PV x (1 + r) n
Present Value Formula:
PV = FV / (1 + r) n

21. Future Value Example
Suppose you purchased a house ten years ago for $125,000 and houses in your neighborhood have increased in value by an average of 10% per year. What should your house be worth today?
FV = PV(1 + r) n
FV = 125,000 (1.10) 10 = 324,218
N = 15; I/Y = 8; PV = 500; CPT FV = -1,586.08
Formula: 500(1.08)15 = 500( ) = 1,586.08
(500)(.08) = 1,100
Lecture Tip: You may wish to take this opportunity to remind students that, since compound growth rates are found using only the beginning and ending values of a series, they convey nothing about the values in between. For example, a firm may state that “EPS has grown at a 10% annually compounded rate over the last decade” in an attempt to impress investors of the quality of earnings. However, this just depends on EPS in year 1 and year 11. For example, if EPS in year 1 = $1, then a “10% annually compounded rate” implies that EPS in year 11 is (1.10)10 = So, the firm could have earned $1 per share 10 years ago, suffered a string of losses, and then earned $2.59 per share this year. Clearly, this is not what is implied by management’s statement above.

22. Example 4.2 Personal Finance Present Value of a Single Sum
You are considering investing in a savings bond that will pay $15,000 in ten years. If the competitive market interest rate is fixed at 6% per year, what is the bond worth today?
PV = FV / (1 + r) n
PV = 15,000 / (1.06) 10 = 8,375.92

23. 4.3 Valuing a Stream of Cash Flows
Previous examples were easy to evaluate because there was one cash flow. What do we need to do if there are multiple cash flow?
Equal Cash Flows: Annuity or Perpetuity
Unequal/Uneven Cash Flows

24. 4.3 Valuing a Stream of Cash Flows?
Uneven cash flows are when there are different cash flow streams each year
Treat each cash flow as a Single Sum problem and add the PV amounts together.

25. 4.3 Valuing a Stream of Cash Flows
What is the present value of the preceding cash flow stream using a 12% discount rate?
PV = FV / (1 + r) n
Yr 1 $1,000 / (1.12)1 = $ 893
Yr 2 $2,000 / (1.12)2 = 1,594
Yr 3 $3,000 / (1.12)3 = 2,135
$4,622
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26. 4.4 The Net Present Value of a Stream of Cash Flows
We can represent investment decisions on a timeline as negative cash flows.
The NPV of an investment opportunity is also the present value of the stream of cash flows of the opportunity:
If the NPV is positive, the benefits exceed the costs and we should make the investment.

27. 4.4 The Net Present Value of a Stream of Cash Flows
You have been offered the following investment opportunity: If you invest $1 million today, you will receive the cash flow stream below. If you could otherwise earn 12% per year on your money, should you undertake the investment opportunity?
Yr 1 $250,000
Yr 2 $150,000
Yr 3 $200,000
Yr 4 $300,000
Yr 5 $400,000
Yr 6 $500,000
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28. 4.5 Annuities and Perpetuities
Annuity: A series of equal cash flows for a specified number of periods at a constant interest rate
Ordinary annuity: cash flows occur at the end of each period
Annuity due: cash flows occur at the beginning of each period (not covered in class)
Perpetuity: An infinite series of equal payments

29. Examples of Annuities and Perpetuities
Annuity:
If you buy a bond, you will receive equal coupon interest payments over the life of the bond.
If you borrow money to buy a house or a car, you will pay a stream of equal payments.
Amortization Schedule
Perpetuity:
If you buy preferred stock, you will receive equal dividend payments forever.
Assuming infinite life of corporation

30. Annuity Formulas in Text
(Eq. 4.5)
(Eq. 4.6)

31. Annuities Formulas - Simplified
This presentation utilizes a simplified version of the formulas that eliminates the 1/r component:
Lecture Tip: The annuity factor approach is a short-cut approach in the process of calculating the present value of multiple cash flows and that it is only applicable to a finite series of level cash flows. Financial calculators have reduced the need for annuity factors, but it may still be useful from a conceptual standpoint to show that the PVIFA is just the sum of the PVIFs across the same time period.
5.31

32. Interest Rates and Time Periods
Keep in mind that payments (C) are not always annual. Thus the interest rates (r) and number of periods (N) need to be adjusted if there is more than 1 payment per year. For example, here are interest rates and number of periods for a 5-year loan at 6% interest:
Annual Payments: N = 5 r = 6.0%
Semi-Annual Payments: N = r = 3.0%
Quarterly Payments: N = r = 1.5%
Monthly Payments: N = r = 0.5%
The students can read the example in the book.
After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?
Note that the difference between the answer here and the one in the book is due to the rounding of the Annuity PV factor in the book.
5.32

33. Present Value Annuity Example
You can afford to pay $632 per month towards a new car. If you can get a loan that charges 1% interest per month and has a 40-month term, how much can you borrow?
The students can read the example in the book.
After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?
Note that the difference between the answer here and the one in the book is due to the rounding of the Annuity PV factor in the book.
5.33

34. Present Value Annuity Example
You can afford to pay $632 per month towards a new car. If you can get a loan that charges 1% interest per month and has a 40-month term, how much can you borrow?
The students can read the example in the book.
After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?
Note that the difference between the answer here and the one in the book is due to the rounding of the Annuity PV factor in the book.
5.34

35. Example 4.8 Personal Finance Retirement Savings Plan Annuity
Ellen is 35 years old, and she has decided it is time to plan seriously for her retirement. At the end of each year until she is 65, she will save $10,000 in a retirement account. If the account earns 10% per year, how much will Ellen have saved at age 65?

36. Example 4.8 Personal Finance Retirement Savings Plan Annuity
Ellen is 35 years old, and she has decided it is time to plan seriously for her retirement. At the end of each year until she is 65, she will save $10,000 in a retirement account. If the account earns 10% per year, how much will Ellen have saved at age 65?

37. Problems
With $10,000 to invest and assuming a 10% interest rate, how much will you have at the end of 5 years?
Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today?
Which of the following is better?
Investing $500 at a 7% annual interest rate for 5 years.
Investing $500 today and receiving $750 in 5 years.
N = 5
PV = -10,000
At 5%, the FV = 10,000 x (1.05)^5 = 12,762.82
At 4.5%, the FV = 10,000 x (1.045) ^ 5 = 12,461.82
The difference is attributable to interest. That difference is 12, – 12, = 301
To double the 10,000:
Ln(20,000/10,000) / ln(1.05) = 14.2 years
Note, the rule of 72 indicates 72/5 = 14 years, approximately.
r = (4,000/1,000)^1/20 – 1 = 7.18%

38. Problems
An investment will provide you with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually?
Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333, over 30 years If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?
You are saving for a new house, and you put $10,000 per year in an account paying 8%. How much will you have at the end of 3 years?
Present value problems:
End of the year: 10 N; 8 I/Y; 100 PMT; CPT PV =
Beginning of the year: PV = $ X 1.08 = $724.69
Future value problems:
10 N; 8 I/Y; -100 PMT; CPT FV = 1,448.66
10N; 8 I/Y; -1,000 PV; -100 PMT; CPT FV = 3,607.58
5.38

39. Present Value of a Perpetuity
(Eq. 4.4)
PV is the value of the perpetuity today
C is the recurring payment
r is the required interest rate (not dividend rate)
If we know any two variables, we can solve for the third

40. Example 4.6 Personal Finance Endowing a Perpetuity
You want to endow an annual graduation party at your alma mater that is budgeted to cost $30,000 per year forever. If the university can earn 8% per year on its investments and the first party is in one year’s time, how much will you need to donate to endow the party?

41. Example 4.6 Personal Finance Endowing a Perpetuity
You want to endow an annual graduation party at your alma mater that is budgeted to cost $30,000 per year forever. If the university can earn 8% per year on its investments and the first party is in one year’s time, how much will you need to donate to endow the party?

42. Perpetuity Example Preferred Stock
There is a preferred stock with a par value of $25 and a dividend rate of 5% per year. If preferred stock with a similar risk profile is yielding 4%, what price should the stock be trading at?

43. Loan Payments and Amortization Schedules
One of the most useful applications of the annuity formula is calculating the payment on a loan. To do so, we revise the annuity formula to solve for Payment (C):
(Eq. 4.8)

44. Example 4.10 Computing a Loan Payment
Your firm plans to buy a warehouse for $100,000. The bank offers you a 30-year loan with equal annual payments and an interest rate of 8% per year. The bank requires that your firm pay 20% of the purchase price as a down payment, so you can borrow only $80,000. What is the annual loan payment?

45. Example 4.10 Computing a Loan Payment
Eq. 4.8 calculates the payment as follows:

46. Loan Payments and Amortization Schedules
In Example 4.10, our firm will need to pay $7, each year to fully repay the loan over 30 years.
To understand how the loan will be repaid, we must create an Amortization Schedule for the loan.

47. Amortization Schedule
An Amortization Schedule contains five columns, as illustrated below:
The payment is typically monthly, quarterly, or annually, so we may need to adjust the interest rate and number of periods accordingly.
Pmt #
Payment
Interest
Principal
Balance
$80,000 1
$7,106
$6,400
$706
$79,294

48. Amortization Schedule: Equal Payments
Stays the Same
Interest Expense
Declines Over Time
Principal Repayment
Increases Over Time
Balance of Loan
Fully repaid by maturity
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49. Amortization Schedule: Equal Payments
Example 4.10:
Purchase Warehouse for $100,000
$20,000 Down Payment; $80,000 Loan
30-year Term
8% Interest Rate
Annual End of Year Payments

50. Amortization Schedule
Pmt #
Payment
Interest
Principal
Balance
$80,000 1
$7,106
Beginning Balance is Loan Amount
Calculate Payment:

51. Amortization Schedule
Pmt #
Payment
Interest
Principal
Balance
$80,000 1
$7,106
$6,400
$706
$79,294
Calculate Interest
Principal x Rate x Time: ($80,000)(8%)(1 yr)
Principal = Payment - Interest
Balance = Prior Balance - Principal

52. Amortization Schedule

53. Amortization Example: Equal Payments
Create an amortization schedule for the first month based on the following terms:
Purchase Building for $1,000,000
$200,000 Down Payment, Balance Borrowed
5 yr. Term
9% Interest Rate
Monthly Payments

54. Calculating Payment in Excel
Excel can easily calculate the payment for a loan: