1. Chapter 4 The Time Value of Money

2. Chapter Outline
4.1 The Timeline 4.2 The Three Rules of Time Travel 4.3 Valuing a Stream of Cash Flows 4.4 Calculating the Net Present Value 4.5 Perpetuities and Annuities

3. Chapter Outline (cont’d)
4.6 Solving Problems with a Spreadsheet or Calculator 4.7 Non-Annual Cash Flows 4.8 Solving for the Cash Payments 4.9 _The Internal Rate of Return

4. 4.1 The Timeline
A timeline is a linear representation of the timing of potential cash flows.
Drawing a timeline of the cash flows will help you visualize the financial problem.

5. 4.1 The Timeline (cont’d) Assume that you made a loan to a friend.
You will be repaid in two payments, one at the end of each year over the next two years.

6. 4.1 The Timeline (cont’d)
Differentiate between two types of cash flows
Inflows are positive cash flows.
Outflows are negative cash flows, which are indicated with a – (minus) sign.

7. 4.1 The Timeline (cont’d)
Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments.
The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow.
Timelines can represent cash flows that take place at the end of any time period – a month, a week, a day, etc.

8. Textbook Example 4.1

9. Textbook Example 4.1 (cont’d)

10. 4.2 Three Rules of Time Travel
Financial decisions often require combining cash flows or comparing values.
Three rules govern these processes.
Table 4.1 The Three Rules of Time Travel

11. The 1st Rule of Time Travel
A dollar today and a dollar in one year are not equivalent.
It is only possible to compare or combine values at the same point in time.
Which would you prefer: A gift of $1,000 today or $1,210 at a later date?
To answer this, you will have to compare the alternatives to decide which is worth more. One factor to consider: How long is “later?”

12. The 2nd Rule of Time Travel
To move a cash flow forward in time, you must compound it.
Suppose you have a choice between receiving $1,000 today or $1,210 in two years.
You believe you can earn 10% on the $1,000 today, but want to know what the $1,000 will be worth in two years.
The time line looks like this:

13. The 2nd Rule of Time Travel (cont’d)
Future Value of a Cash Flow

14. The 2nd Rule of Time Travel (cont’d)
The following graph shows the account balance and the composition of interest over time when an investor starts with an initial deposit of $1000, shown in red, in an account earning 10% interest over a 20-year period.
Note that the turquoise area representing interest on interest grows, and by year 15 has become larger than the interest on the original deposit, shown in green.
In year 20, the interest on interest the investor earned is $ , while the total interest earned on the original $1000 is $2000.

15. Figure 4.1 The Composition of Interest Over Time

16. Textbook Example 4.2

17. Textbook Example 4.2 (cont’d)

18. Alternative Example 4.2 Problem
Suppose you have a choice between receiving $5,000 today or $10,000 in five years.
You believe you can earn 10% on the $5,000 today, but want to know what the $5,000 will be worth in five years.

19. Alternative Example 4.2 (cont’d)
Solution
The time line looks like this:
In five years, the $5,000 will grow to:
$5,000 × (1.10)5 = $8,053
The future value of $5,000 at 10% for five years is $8,053.
You would be better off forgoing the gift of $5,000 today and taking the $10,000 in five years.

20. The 3rd Rule of Time Travel
To move a cash flow backward in time, we must discount it.
Present Value of a Cash Flow

21. Textbook Example 4.3

22. Textbook Example 4.3

23. Alternative Example 4.3 Problem
Suppose you are offered an investment that pays $10,000 in five years.
If you expect to earn a 10% return, what is the value of this investment today?

24. Alternative Example 4.3 (cont’d)
Solution
The $10,000 is worth:
$10,000 ÷ (1.10)5 = $6,209

25. Applying the Rules of Time Travel
Recall the 1st rule: It is only possible to compare or combine values at the same point in time.
So far we’ve only looked at comparing.
Suppose we plan to save $1000 today, and $1000 at the end of each of the next two years.
If we can earn a fixed 10% interest rate on our savings, how much will we have three years from today?

26. Applying the Rules of Time Travel (cont'd)
The time line would look like this:

27. Applying the Rules of Time Travel (cont'd)

28. Applying the Rules of Time Travel (cont'd)

29. Applying the Rules of Time Travel (cont'd)

30. Table 4.1 The Three Rules of Time Travel

31. Textbook Example 4.4

32. Textbook Example 4.4 (cont’d)

33. 4.3 Valuing a Stream of Cash Flows
Based on the first rule of time travel we can derive a general formula for valuing a stream of cash flows
If we want to find the present value of a stream of cash flows, we simply add up the present values of each.

34. 4.3 Valuing a Stream of Cash Flows (cont’d)
Present Value of a Cash Flow Stream

35. Textbook Example 4.5

37. Future Value of Cash Flow Stream
Future Value of a Cash Flow Stream with a Present Value of PV

38. 4.4 Calculating the Net Present Value
Calculating the NPV of future cash flows allows us to evaluate an investment decision.
Net Present Value compares the present value of cash inflows (benefits) to the present value of cash outflows (costs).

39. Textbook Example 4.6

41. 4.5 Perpetuities and Annuities
When a constant cash flow will occur at regular intervals forever it is called a perpetuity.

42. 4.5 Perpetuities and Annuities (cont’d)
The value of a perpetuity is simply the cash flow divided by the interest rate.
Present Value of a Perpetuity

43. Textbook Example 4.7

44. Textbook Example 4.7 (cont’d)

45. 4.5 Perpetuities and Annuities (cont’d)
When a constant cash flow will occur at regular intervals for a finite number of N periods, it is called an annuity.
Present Value of an Annuity

46. Present Value of an Annuity
To find a simpler formula, suppose you invest $100 in a bank account paying 5% interest.
As with the perpetuity, suppose you withdraw the interest each year.
Instead of leaving the $100 in forever, you close the account and withdraw the principal in 20 years.

47. Present Value of an Annuity (cont’d)
You have created a 20-year annuity of $5 per year, plus you will receive your $100 back in 20 years. So:
Re-arranging terms:

48. Present Value of an Annuity
For the general formula, substitute P for the principal value and:

49. Future Value of an Annuity

50. Textbook Example 4.9

51. Textbook Example 4.9 (cont’d)

52. Growing Cash Flows Growing Perpetuity
Assume you expect the amount of your perpetual payment to increase at a constant rate, g.
Present Value of a Growing Perpetuity

53. Textbook Example 4.10

54. Textbook Example 4.10 (cont’d)

55. Growing Cash Flows Growing Annuity
The present value of a growing annuity with the initial cash flow c, growth rate g, and interest rate r is defined as:
Present Value of a Growing Annuity

56. Textbook Example 4.11

58. Textbook Example 4.12

59. Textbook Example 4.12 (cont’d)

60. Textbook Example 4.13

61. Textbook Example 4.13 (cont’d)

62. 4.9 The Internal Rate of Return
In some situations, you know the present value and cash flows of an investment opportunity but you do not know the internal rate of return (IRR), the interest rate that sets the net present value of the cash flows equal to zero.

63. Textbook Example 4.16

64. Textbook Example 4.16 (cont’d)

65. Chapter 5 Interest Rates

66. Chapter Outline
5.1 Interest Rate Quotes and Adjustments 5.2 Application: Discount Rates and Loans 5.3 The Determinants of Interest Rates 5.4 Risk and Taxes 5.5 The Opportunity Cost of Capital

67. 5.1 Interest Rate Quotes and Adjustments
How do we determine the interest rate?
In practice, interest is paid and interest rates are quoted in different ways.
For example, in mid-2012, Metropolitan National Bank offered savings accounts with an interest rate of 1.65% paid at the end of each year, while AIG Bank offered an annual interest rate of only 1.60%, but paid on a daily basis.
Interest rates can also differ depending on the investment horizon.
In July 2012, investors earned less than 0.25% on one-year risk-free U.S. Treasury Bills, but could earn more than 2.3% on twenty-year Treasuries.

68. 5.1 Interest Rate Quotes and Adjustments
Interest rates are often stated as an effective annual rate (EAR), which indicates the actual amount of interest that will be earned at the end of one year
The Effective Annual Rate
Indicates the total amount of interest that will be earned at the end of one year
Considers the effect of compounding
Also referred to as the effective annual yield (EAY) or
Annual percentage yield (APY)

69. 5.1 Interest Rate Quotes and Adjustments (cont'd)
Adjusting the Discount Rate to Different Time Periods
Earning a 5% return annually is not the same as earning 2.5% every six months.
In general, we can convert a discount rate of r for one period to an equivalent discount rate for n periods using the following formula:
General Equation for Discount Rate Period Conversion
For example, convert the 5% return annually to the equivalent for every six months:
(1.05)0.5 – 1= – 1 = = 2.47%
n = 0.5 since we are solving for the six month (or 1/2 year) rate

70. Textbook Example 5.1

72. 5.1 Interest Rate Quotes and Adjustments (cont'd)
Banks also quote interest rates in terms of an annual percentage rate (APR), which indicates the amount of simple interest earned in one year, that is, the amount of interest earned without the effect of compounding.
Because it does not include the effect of compounding, the APR quote is typically less than the actual amount of interest that you will earn.
To compute the actual amount that you will earn in one year, we must first convert the APR to an effective annual rate.

73. Annual Percentage Rates
The annual percentage rate (APR), indicates the amount of simple interest earned in one year.
Simple interest is the amount of interest earned without the effect of compounding.
The APR is typically less than the effective annual rate (EAR).

74. Annual Percentage Rates (cont'd)
The APR itself cannot be used as a discount rate.
The APR with k compounding periods is a way of quoting the actual interest earned each compounding period:

75. Annual Percentage Rates (cont'd)
Converting an APR to an EAR
The EAR increases with the frequency of compounding.
Continuous compounding is compounding every instant.

76. Annual Percentage Rates (cont'd)
Table 5.1 Effective Annual Rates for a 6% APR with Different Compounding Periods
A 6% APR with continuous compounding results in an EAR of approximately %.

77. Textbook Example 5.2

79. Alternative Example 5.2 Problem
A firm is considering purchasing or leasing a luxury automobile for the CEO. The vehicle is expected to last 3 years.
You can buy the car for $65,000 up front , or you can lease it for $1,800 per month for 36 months.
The firm can borrow at an interest rate of 8% APR with quarterly compounding.
Should you purchase the system outright or pay $1,800 per month?

80. Alternative Example 5.2 (cont’d)
Solution
The first step is to compute the discount rate that corresponds to monthly compounding.
To convert an 8% rate compounded quarterly to a monthly discount rate, compound the quarterly rate using Equations 5.3 and 5.1:

81. Alternative Example 5.2 (cont’d)
Solution
Given a monthly discount rate of %, the present value of the 36 monthly payments can be computed:
Paying $1,800 per month for 36 months is equivalent to paying $57,486 today. This is $65,000 - $57,486 = $7,514 lower than the cost of purchasing the system, so it is better to lease the vehicle rather than buy it.

82. 5.2 Application: Discount Rates and Loans
Computing Loan Payments
Payments are made at a set interval, typically monthly.
Each payment made includes the interest on the loan plus some part of the loan balance.
All payments are equal and the loan is fully repaid with the final payment.

83. 5.3 The Determinants of Interest Rates
The interest rates that are quoted by banks and other financial institutions, and that we have used for discounting cash flows, are nominal interest rates, which indicate the rate at which your money will grow if invested for a certain period.
Of course, if prices in the economy are also growing due to inflation, the nominal interest rate does not represent the increase in purchasing power that will result from investing.
The rate of growth of your purchasing power, after adjusting for inflation, is determined by the real interest rate

84. 5.3 The Determinants of Interest Rates
Inflation and Real Versus Nominal Rates
Nominal Interest Rate: The rates quoted by financial institutions and used for discounting or compounding cash flows
Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation

85. 5.3 The Determinants of Interest Rates (cont'd)
The Real Interest Rate

86. Textbook Example 5.4

87. Textbook Example 5.4 (cont'd)

88. Figure 5.1 U.S. Interest Rates and Inflation Rates,1960–2012

89. Investment and Interest Rate Policy
An increase in interest rates will typically reduce the NPV of an investment.
Consider an investment that requires an initial investment of $10 million and generates a cash flow of $3 million per year for four years. If the interest rate is 5%, the investment has an NPV of:

90. Investment and Interest Rate Policy (cont'd)
If the interest rate rises to 9%, the NPV becomes negative and the investment is no longer profitable:

91. Monetary Policy, Deflation, and the 2008 Financial Crisis
When the 2008 financial crisis struck, the Federal Reserve responded by cutting its short-term interest rate target to 0%.
While this use of monetary policy is generally quite effective, because consumer prices were falling in late 2008, the inflation rate was negative, and so even with a 0% nominal interest rate the real interest rate remained positive.

92. The Yield Curve and Discount Rates
You may have noticed that the interest rates that banks offer on investments or charge on loans depend on the horizon, or term, of the investment or loan.
The relationship between the investment term and the interest rate is called the term structure of interest rates.
We can plot this relationship on a graph called the yield curve.

93. Figure 5. 2 Term Structure of Risk-Free U. S
Figure 5.2 Term Structure of Risk-Free U.S. Interest Rates, November 2006, 2007, and 2008

94. The Yield Curve and Discount Rates (cont'd)
The term structure can be used to compute the present and future values of a risk-free cash flow over different investment horizons.
Present Value of a Cash Flow Stream Using a Term Structure of Discount Rates

95. Textbook Example 5.5

96. Textbook Example 5.5 (cont'd)

97. The Yield Curve and the Economy
Interest Determination
The Federal Reserve determines very short-term interest rates through its influence on the federal funds rate, which is the rate at which banks can borrow cash reserves on an overnight basis.
All other interest rates on the yield curve are set in the market and are adjusted until the supply of lending matches the demand for borrowing at each loan term.

98. The Yield Curve and the Economy
Interest Rate Expectations
The shape of the yield curve is influenced by interest rate expectations.
An inverted yield curve indicates that interest rates are expected to decline in the future.
Because interest rates tend to fall in response to an economic slowdown, an inverted yield curve is often interpreted as a negative forecast for economic growth.
Each of the last six recessions in the United States was preceded by a period in which the yield curve was inverted.
The yield curve tends to be sharply increasing as the economy comes out of a recession and interest rates are expected to rise.

99. Figure 5. 3 Short-Term Versus Long-Term U. S
Figure 5.3 Short-Term Versus Long-Term U.S. Interest Rates and Recessions

100. 5.4 Risk and Taxes Risk and Interest Rates
U.S. Treasury securities are considered “risk- free.”
All other borrowers have some risk of default, so investors require a higher rate of return.

101. Figure 5.4 Interest Rates on Five- Year Loans for Various Borrowers, July 2012
Source: FINRA.org.

102. Textbook Example 5.7

103. Textbook Example 5.7 (cont'd)

104. After-Tax Interest Rates
If the cash flows from an investment are taxed, the investor’s actual cash flow will be reduced by the amount of the tax payments.
Here, we consider the effect of taxes on the interest earned on savings (or paid on borrowing).
Taxes reduce the amount of interest an investor can keep, and we refer to this reduced amount as the after-tax interest rate.

105. Textbook Example 5.8

107. Continuous Rates and Cash Flows
Discount Rate of a Continuously Compounded APR
Some investments compound more frequently than daily.
As we move from daily to hourly to compounding every second, we approach the limit of continuous compounding, in which we compound every instant.
The EAR for a Continuously Compounded APR

108. Continuous Rates and Cash Flows (cont’d)
Discount Rate of a Continuously Compounded APR
Alternatively, if we know the EAR and want to find the corresponding continuously compounded APR, the formula is:
The Continuously Compounded APR for an EAR
Continuously compounded rates are not often used in practice.